EXERCISE - 1
A)(2√2,π/4) B)(2√2,3π/4) C) (2√2, -3π/4) D) (2√2,-π/4)
2) The polar coordinates of the point whose cartesian co-ordinate (-√3,1), are
A) (2,2π/3) B)(2,5π/6) C) (2,-5π/6) D) (2,-2π/3)
3) the Cartesian coordinates of the point whose polar coordinates are (√3,-3π/4), are
A) (√6/2,-√6/2) B) (-√6/2,√6/2) C) (-√6/2,-√6/2) D) none
4) the polar coordinates of the centroid of the Triangle formed by the points A(3,2), B(-6,-3), C(0,-2) are
A)(√2,π/4) B) (√2,-π/4) C)(√2,3π/4) D) (√2, -3π/4)
5) The Cartesian coordinates of A and the polar coordinates of B are respectively A(2,3) and (2,60°). the coordinates of the point at which AB is divided internally in the ratio 2:1 are
A) (4/3, √3(2+√3)/3)
B) (5/3, √3(2+√3)/3)
C)(4/3, √3) D) (5/3, √3)
6) If the coordinates of the centroid of the Triangle formed by the points then A(2x,3x), B(y,2y), C(-1,-3) are (2,3), then coordinates of A are
A) (6,6) B)(9,4) C)(4,9), D) none
7) The points (0,-2) (2,4),(-1,-5), form
A) an isosceles triangle
B) right angled triangle
C) an equilateral triangle D) n
8) the point (1,2) divides the line segment AB joining the points A(3,2) and then B(-2,2) internally in the ratio k:1. Then k is equal to
A) 2/3 B) 3/2 C)1/2, D) 2
9) The x-axis divides the line segment AB, where A=(2,-3), B=(5,6), in the ratio
A) 2:3 B)1:2 C) 2:1 D) 3:2
10) the coordinates of two points A and B are (3,-3) and (-5,7) respectively in the line y= x divides AB in the ratio
A) 2:2 B)1:2 C) 2:3 D) 3:2
11) A(1,6), B(3, -4), C(x,y) are the three collinear points such that AB= BC, then the value of x and y satisfy.
A) x²+ y²= 220. B) 5x+ 14y= 0.
C) 13x+ 5y+5=0. D) none
12) The extremities of the diagonal are (3,-4) and (-6,5). If the third vertex is the point (-2,1), then the coordinates of the fourth vertex are.
A)(-1,0) B)(1,0) C)(0,1), D) (0,-1)
13) The coordinates of the points A, B, C are (-1,-1),(5,7) and (1,-15). The length of the median through A is..
A)√41 B) √29 C)5 D) none
14) the coordinates of the orthocentre of the Triangle, formed by the straight lines given by (x-2)(y-2)(x+y-1)= 0, are
A)(2,-1) B)(-1,2) C)(2,2), D)(3/2,3/2)
15) the coordinates of the four points A,B,C,D are (0,0),(0,10),(8,16), (8,6) respectively, if the points are joined in order, then which one is the most appropriate statement ?
A) ABCD forms a square
B) ABCD forms Rhombus
C) ABCD forms a parallelogram
D) none of these statements is true
16) The points (2a,0), (0,2b), (1,1) are collinear if
A) 1/a + 1/b= 2
B) 2/a + 2/b= 1
C) 1/a + 1/b= 1. D) none
17) if two vertices of an equilateral triangle have retional coordinates, then for the third vertex which one is most applicable
A) the coordinates are integral
B) the coordinates are rational
C) the coordinates are irrational
D) at least one coordinate is irrational.
18) If the coordinates of the midpoints of the sides of a triangle ABC are (0,0),(2,-1),(-1,3), then the coordinates of the centroid of ∆ABC are
A) (1/3,2/3) B) (-4,-3). C) (-3,-4) D) none
19) the coordinates of the three vertices of the triangle ABC are (-2,1), (-1,-3), (3,-2); then the coordinates of its circumcentre are
A) (0,0) B)(0,-4). C)(0,-4/3) D) n
20) The equations of the three sides of a triangle are x-2y+4=0, 2x +y- 7=0 , x + y+3=0. the coordinates of the orthocentre of the triangle are
A) (3/2,17/12) B)(2,3) C)(3,2) D) n
21) The area of a triangle is 5 square unit. Two of the vertices are (2,1),(3,-2); the third vertex lies on the line y= x+3. The coordinates of the third vertex are
A) (7/2,13/2),(3/2,-3/2)
B) (7/2,13/2),(-3/2,3/2)
C) (-7/2,13/2),(-3/2,3/2) D) n
22) The coordinates of two vertices of a triangle are (2,2),(3,1), the third vertex lies on the line y+ 3x= 0. If the coordinates of the centroid of the triangle are (2,0), then the coordinates of the third vertex are
A) (1/3,-1), B)(-1,3) C)(1,-3) D) n
23) the coordinates of the three vertices of a triangle are (2,7),(5,1),(x,3) and the area of the triangle is 18 sq. unit. The value/s of x is (are)
A) 10 B) 2, -10 C) 10, -2 D) n
24) the area of the quadrilateral whose vertices are (a,0)(-b,0),(0, a), (0,-b) (with a,b > 0) is
A) 0 B) (a+b)²/2 C) (a²+b²+ab)/2 D) n
25) If the coordinates of the points A,B,C,D are (6,3),(-3,5),(4,-2) and (x,3x) respectively, and (area of∆DBC)/(area of ∆ABC) = 1/2, then the value of x is
A) 8/11 B) 11/8 C) 3/11 D) 11/3
26) the circumcenter of the triangle formed by the point (-3,1),(1,3) and (3,0) lies on 2x+y= 0. the coordinates of the circumcenter are
A) (1/16,-1/8) B) (-1/8,1/4) C) (-1/16,1/8) D) (1/8,-1/4)
27) A rectangle has two opposite vertices at the points (1,2) and (5,5). If the other vertices lie on the line x= 3, then the coordinates of the other vertices are
A) (3,2),(3,6) B)(3,1),(3,6)
C) (3,1),(3,5) D)(3,-1),(3,-6)
28) The locus of the centroid of the triangle whose vertices are (a cosk, b sink), (a sink, - b cosk) and (1,2), where k is a parameter, is
A) {(3x+1)/a}²+ {(3x+2)/b}²=2
B) (3x-1)²+ (3x-2)²=a²- b²
C) (3x+1)²+ (3x+2)²=a²- b²
D) {(3x-1)/a}²+ {(3x-2)/b}²=2.
29) The locus of centroid of a triangle whose vertices are (1,0),(a sec k, a tan k) and (b sec k, - b tank), where k is a parameter, is
A) (3x-1)²/(a+b)²- 9y²/(a-b)²= 1
B) (3x-1)² - 9y²=(a+b)²
C) (3x+1)²/(a+b)²- 9y²/(a-b)²= 1
D) (3x+1)²/(a+b)²- 9y²= a²-b²
30) If the equation of the locus of a points (m,n) and (s,t) is (m-s)x + (n- t)y + c= 0, then the value of c is
A) 1/2 (m²+n²+ s²+ t²)
B)1/2 (s²+t²- m²-n²)
C) 1/2 (m²-s²+ n²- t²)
D) √{(m²+n²- s²- t²)}
31) The locus of the point which divides the line segment joining the points (a,0) and (0,b) in the ratio 2:3 is
A) 2bx- 2ay= 0 B) 3bx+ 2ay= 0
C) 2bx- 3ay= 0 D)2bx + 3ay= 0
32) The locus of the point which divides the line segment joining the points (a cos t,0) and (0,b sin t) in the ratio m:n is
A) x²/a²n² + y²/b²m²= 1/(m+n)²
B) x²/a²m² + y²/b²n²= 1/(m+n)²
C) x²/b²n² + y²/a²m²= 1/(m+n)²
D) none
33) A(a cos k, a sin k) and B (b cos t, b sin t) are two points M(x,y) is another point such M divides the line segment AB internally in the ratio a: b, then tan{(k+t)/2} is equal to
A)x/y B)y/x C)(x+y)/(x-y) D)-x/y
34) A(a cos k, a sin k) and B (b cos t, b sin t) are two points. M(x,y) is another point such that M divides the line segment AB externally in the ratio a: b, then tan{(k+t)/2} is equal to
A)x/y B)y/x C)(x+y)/(x-y) D)-x/y
35) The locus of the midpoint of the portion of the line x cos k+ y sin k= p intercepted between the axes is
A) x²+ y²= 4/p²
B) x²+ y²= p²/4
C) 1/x²+ 1/y²= p²/4
D) 1/x²+ y²= 4/p²
36) the coordinates of the points at a distance 2√2 unit from the (2,3) in the direction making an angle 45° with the positive direction of the x-axis, are
A) (0,1),(4,1) B) (0,5),(4,5)
C) (4,5),(0,1). D) (0,5),(4,1)
37) If the area of the triangle formed by x cos k + y sin k= p with the co-ordinate Axes is always k², then the locus of the midpoint of the segment of the line intercepted between axes is
A) 2xy=± k²/p² B) xy=± 2k²/p²
C) 2xy=± k² D) xy=± 2k²
38) The parametric coordinates of a point P are {(t+1)/(t-1), 2t+3}, where t is a parameter, then the locus of P is
A) x(y-3)= y-1. B) x(y-5)= y-1
C) x(y-5)= y-3 D) none
39)If k be a parameter, then the locus of the point P{2/(1+ sin k), 3 cos k} is
A) x²y²= 36(x-1) B)x²y²= 36(x+1)
C) x²y²= -36(x-1) D) x²y²= - 36(x-1)
40) If t be a parameter, then the locus of the point P(2t - 3/t, 2t+3/t) is
A) x²-y²= 6 B) y²-x²= 6
C) x²-y²= 24 D) y²-x²= 24
41) Let P(1,2) and Q(3,4) be two points. The point R on the x-axis is such that PR+ RQ is minimum. The coordinates of R are
A) (3/5,0) B) (5/3,0) C) (-3/5,0) D) (-5/3,0)
42) The Four Points (-a, -b),(0,0),(a,b) and (a², ab) form a
A) parallelogram
B) a square
C) a quadrilateral area ab/2 √(a²+ b²) D) none
43) the base of a triangle lies along the line x= a and is of length a. if the area the triangle is a², then its vertex lies on the line.
A) x= -2a. B)x= 2a C) x= 3a D) x= -3a
44) The locus of the point (2t²+t+1, t²- t+1) is
A) (x-2y+1)²=3(x+y-2)
B) (x-2y+1)²=3(x+y+2)
C) (x-2y+1)²=3(x-y+2)
D) (x-2y+1)²=3(x- y-2)
45) The polar equation of y= tan k with respect to the origin as pole and the +ve y-axis as the initial line is...
A) $= k B) $= π/2 - k C) $= - k D) n
46) The cartesian equation of r²= a² cos 2k is.
A) (x²-y²)² = a²(x²+ y²)
B) (x²+y²)/(x²- y²) = a²
C) (x² + y²)² = a²(x² - y²)
D)(x²-y²)/(x²+y²) = a²
47) the cartesian equation of k= t is
A) y= x sin t B) y= x cos t
C) y= x tan t D) x= y tan t
48) the cartesian equation of r cos²(t/2)= 1 is
A) y²= 4(1-x). B)y²= 4(x -1)
C) x²= 4(1-y) D) x²= 4(y- 2)
49) The Cartesian equation of √r = √a cos (t/2) is.
A) (2x²+2y²+ax)²= a²(x²+y²)
B) (2x²+2y²- ax)²= a²(x²+y²)
C) (2x²+2y²+ax)²= a²(x²-y²)
D) (2x²+2y²-ax)²= a²(x²-y²)
50) The coordinates of the points A, B, C, P are (6,3),(-3,5),(4,-2) and (x,y) respectively; then (Area of∆PBC)/(Area of ∆ABC) is equal to.
A) |(x+y+2)/7| B) |(x-y+2)/7|
C)(x+y-2)/7| D) none
51) the new coordinates of the point (4,3) when the co-ordinate Axes are translated by shifting the origin to (-2,1) are
A)(6,2) B)(2,4) C)(6,4) D)(2,2)
52) without changing the direction of the axes, the origin is transferred to the point (-4,-7). the coordinates of point P in the new system, are (5,-2). The coordinates of P in the original system, are
A)(9,5 B)(1,5) C)(9,-9) D)(1,-9)
53) Let P be the image of the point (2,-3) with the respect to the x-axis. Then the coordinates of the point P in the new system of coordinates obtained by translation of axes in which the origin is shifted to (-3, 2), are
A) (1,1) B)(-1,5) C)(5,1) D)(-1,1)
54) the co-ordinates Axes are translated by shifting the origin to the point (-3,4). in the new system of co-ordinate Axes, the respective x and y intercepts of a straight line l of which the equation in the original system is 2x+3y= 5, are
A) -1/2, -1/3 B) 11/2, 11/3
C) 23/2, 23/3 D) none
55) By a translation of Axes if the origin be transferred to (a,b), so that the linear terms in the equation (x+y)(x-y-2)=4 are eliminated, then the point (a,b) is.
A) (1,-1) B)(-1,1) C)(-1,-1) D)(1,1)
56) If the origin. (0,0) is shifted to the point (2,3), by a translation of axes, the equation x²+y² - 4x - 6y+9= 0 changes to.
A) x²+y² +4= 0 B) x²+y² -4= 0
C) x²+y² - 8x - 12y+ 48 = 0 D) n
EXERCISE - 2
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1) The points (-a,-b),(0 0),(a,b) and (a²,ab) are
A) collinear
B) vertices of a parallelogram
C) vertices of a rectangle D) n
2) The points (0,8/3),(1,3),(82,30) are vertices of
A) an obtuse angled Triangle
B) an acute angled triangle
C) right angled triangle
D) isosceles triangle E) none
3) Let P and Q be points on the line joining A(-2,5) and B(3,1) such that AP= PQ= QB. Then the midpoint of PQ is
A) (1/2,3) B) A(-1/2,4) C)A(2,3) D) A(-1,4)
4) length of the median from B on AC where A(-1,3), B(1,-1) C(5,1) is
A) √18 B) √10 C) 2√3 D) 4
5) The points (0,-1),(-2,3),(6,7) and (8,3) are
A) collinear
B) vertices of a parallelogram which is not a rectangle
C) vertices of a rectangle is not a square D) Square
6) If (3,-4) and (-6,5) are the extremities of the diagonal of a parallelogram and (-2,1) is its third vertex, then its fourth vertex is
A) (-1,0) B) (0,-1) C) A(-1,1) D) n
7) if O be the origin and if P(a,b) and Q(c,d) be two points, then IP. OQ cos (ang.POQ) is equals to
A) ad+ cb B) (a²+ b²)(c²+d²)
C) (a-c)²+ (b-d)² D) ac+ bd
8) If A and B are two points having coordinates (3,4) and (5,-2) respectively and P is a point such that PA= PB and area of triangle PAB=10 square units, then the coordinates of P are
A)(7,4) or(13,2) B)(7,2) or(13,4)
C)(2,7) or(4,13) D) none
9) Let O be the origin and A, B be two points having coordinates (0,4) or(6,0) respectively. If point P moves in such a way that the area of ∆POA is always twice the area of∆POB. Then P lies on
A) y²= 9x² B) x²= -9y² C) y²+ 9x²=0 D) none
10) The coordinates of the middle points of the sides of a triangle are (4,2),(3,3) and (2,2), then the coordinates of its centroid are
A)(3, 7/3) B) (3,3) C) (4,3) D)n
11) The incentre of the triangle whose vertices are (-36,7),(20,7) and (0,-8) is
A) (0,-1) B(-1,0) C)(1/2,1) D) n
12) the coordinates of the third vertex of an equilateral triangle whose two vertices are (3,4) and (-2,3) are
A) (1,1) or (1,-1)
B)((1+√3)/2), ((7-5√3)/2)) or ((1-√3)/2), ((7+5√3)/2)
C) (-√3,√3) or (√3,-√3) D) n
13) the triangle with vertices at (2,4),(2,6) and (2+√3,5) is
A) right angled
B) right angled isosceles
C) equilateral
D) obtuse angled
14) The area of the triangle with vertices at the point (a, b+c),(b, c+a), (c, a+b) is
A) 0 B) a+b+c C) ab+bc+ ca D) n
15) If P(1,0), Q(-1,0) and R(2,0) are three given points, then the locus of point S satisfying the relation SQ²+ SR²= 2SP² is
A) a straight line parallel to x-axis
B) circle with centre at the origin
D) straight line parallel to y axis
16) the locus of the midpoint of the portion intercepted between the axes by the line x cos k + y sin k= p, where p is a constant is
A) x²+ y²= 4p².
B) 1/x²+ 1/y²= 4/p²
C) x²+ y²= 4/p²
D) 1/x²+ 1/y²= 2p²
17) the locus of the point of intersection of lines x cos k + y sin k= a and x sink - y cos k = b is (k is a variable)
A) 2(x²+ y²= (a²+ b²)
B) x² - y²= a² - b²
C) x²+ y²= a²+ b²
D) none
18) the position of a moving point in x - y plane at time t is given by (u cos k - t, u sin k - gt²/2), where u, k, g are constants. the locus of the moving point is
A) circle B) a parabola
C) an ellipse D) none
19) If A(cos k ,sin k), B(sin k -cos k), C(1,2) are the vertices of the locus of a ∆ABC, then as k varies the locus of its centroid is
A) x²+ y²-2x- 4y+1= 0
B) 3(x²+ y²)-2x- 4y+1= 0
C) x²+ y²-2x- 4y+3= 0 D) none
20) If A and B are two fixed points, then the locus of a point which moves in such a way that the angle APB is a right angle is
A) a circle B) an ellipse
C) a parabola D) none
21) if a variable line passes through the point of intersection of the lines x+2y-1= 0 and 2x-y -1 = 0 and meets the co-ordinate Axes in A and B, then the locus of the midpoint of AB is
A) x+3y= 0 B) x+3y= 10
C) x+3y= 10xy D) none
22) If a variable nearest point on the line drawn through the point of intersection of straight lines x/a+y/b= 1 and x/b +y/a= 1 meets the coordinates axes in A and B, then the locus of the midpoint of AB is
A) ab(x+y)= xy(a+b)
B) ab(x+y)= 2xy(a+b)
C) (a+b)(x+y)= 2abxy D) none
23) The nearest point on the line 3x -4y)= 25 from the origin is
A) (-4,5) B) (3,-4) C) (3,4) D)(3,5)
24) the straight lines x+y= 0, 3x+y= 4, x+3y -4 = 0 form a triangle which is
A) isosceles B) equilateral
C) right angled D) none
25) the image of the point (-1,3) by the line x-y= 0, is
A) (3,-1) B) (1,-3) C) (-1,-1) D)(3,3)
26) If A(1,1), B(√3+1,2) and C(√3, √3 +2) be three vertices of a square, then the diagonal through B is.
A) y= (√3-2)x + (3-√3) B) y= 0
C) x= y D) none
27) If three vertices of a rhombus taken in order are (2,-1),(3,4) and (-2,3), then the fourth vertex is
A) (-3,-2) B) (3,2) C) (2,3) D)(1,2)
28) If (-4,0) and (1,-1) are two vertices of a triangle of area 4 square units, then its third vertex lies on
A) y= x B) 5x+y+12= 0
C) x+5 y -4=0 D) none
29) x₁ y₁ 1 a₁ b₁ 1
If x₂ y₂ 1 = a₂ b₂ 1
x₃ y₃ 1 a₃ b₃ 1 then the two triangles with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) and (a₁,b₁), (a₂,b₂),(a₃, b₃) are
A) equal in area B) similar
C) congruent D) none
30) considered the equation y- y₁ = m(x -x₁). In this equation, if m and x₁ are fixed and different lines are drawn different values of y₁, then
A) the lines will pass through a single point.
B) there will be one possible line only
C) there will be a set of parallel lines
D) none
31) if the sum of the distances is of a point from two perpendicular lines in a plane is 1, then its locus is
A) a square B) circle
C) a straight line
D) two intersecting lines.
32) three lines are px+qy+r= 0, qx+ry+p= 0, and rx+py+1= 0 are concurrent if
A) p+q+r= 0,.
B)p²+q²+r²= PQ+qr+rp
C) p³+q³+r³= 3pq D) none
33) The equation of the line with gradient -3/2 which is concurrent with the lines 4x+3y-7= 0 and 8x+5y-1= 0 is
A) 3x+2y-2= 0 B) 3x+3y-63= 0
C) -3x+2y-2= 0 D) none
34) the point of intersection of the lines x/a +y/b = 1 and x/b +y/a = 1 lies on the line.
A) x -y = 0 B) (x +y)(a+b) = 2ab
C) (lx +my)(a+b) = (l+m)ab
D) (lx - my)(a - b) = (1-m)ab
35) The medians AD and BE of the triangle whose vertices are A(0,b) B(0,0) and C(a,0) are mutually perpendicular if.
A) b=√2 a B) a=√2 b C) b=- √2 a D) a= - √2 b
36) If the equation of the locus of a point equidistant from the points (a₁,b₁) and (a₂ ,b₂) is (a₁- a₂)x + (b₁-b₂)y+c= 0 then the value of c is
A) a₁²- a₂²+b₁² - b₂².
B)√(a₁²- b₁²- a₂²- b₂²)
C) 1/2( a₁²+a₂²+b₁² + b₂²)
D) 1/2(a₂²+b₁² -a₁²- b₂²)
37) area of a triangle with vertices (a,b),(x₁,y₁) and (x₂, y₂), where a₁,x₁ and x₂ are in GP with common ratio r, and , y₁ and y₂ are in GP with common ratio s, is given by
A) ab(r-1)(s-1)(s-r)
B) ab/2 (r+1)(s+1)(s-r)
C) ab/2 (r-1)(s-1)(s-r)
D) ab(r+1)(s+1)(r-s)
38) the equation of perpendicular bisectors of the sides AB and AC of a triangle ABC are x-y+5= 0 and x +2y= 0 respectively. if the point A is (1,-2), the equation of the line BC is
A) 23x+ 14y -40= 0
B)23x+ 14y +40= 0
C) 14x+ 23y -40= 0
D) 14x+ 23y + 40= 0
39) If each of the points (a,4),(-2,b) lies on the line joining the points (2,-1),(5,-3), than the point P(a,b) lies on the line
A) 6(x+y)-25= 0
B) 2x+ 6y +1= 0
C) 2x+ 3y -6= 0
D) 6(x+ y) +25= 0
40) the equation of a line which passes through (a cos³k, a sin³k) and perpendicular to the line x sec k + y cosec k = a is
A) x cos k + y sin k = 2a cos 2k
B) x sin k - y cos k = 2a sin 2k
C) x sin k + y cos k = 2a cos 2k
D) none
41) the ends of the base of an isosceles triangle are at (2a,0) and (0,a). the equation of one side is x= 2a. the equation of the other side is
A) x+ 2y-a= 0 B) x+ 2y= 2a
C) 3x+ 4y- 4a= 0 D) 3x- 4y +4a= 0
42) if the lines and x+ ay +a= 0, bx+ y + b= 0 and cx+ cy +1= 0(a,b,c being distinct ≠1) are concurrent, then the value of a/(a-1) + b/(b-1)+ c/(c-1) is
A) -1 B) 0 C) 1 D) none
43) the set of lines ax+ by +c= 0, Where 3a+ 2b +4c= 0, is concurrent at the point
A) (3/4,1/2). B) (1/2,3/4)
C) (-3/4,-1/2) D) none
44) if a ,b ,c are in AP, then ax+ by +c= 0, represents
A) a single line
B) a family of current lines
C) a family of parallel lines D) n
45) given four lines whose equations are x+ 2y -3= 0, 2x+ 3y -4= 0, 3x+ 4y -7 = 0, and 4x+ 5y -6= 0, then the lines are
A) concurrent
B) sides of a square
C) sides of a rhombus D) n
46) all points lying inside the Triangle formed by the points (1,3), (5,0) and (-1,2) satisfy
A) 3x+ 2y ≥ 0, B) 2x+ y-13≥ 0
C) 2x -3y-12≤ 0 D) -2x+ y≥ 0
47) the distance of the point (3,5) from the line 2x+ 3y-14= 0 measured parallel to the line x- 2y = 1 is
A) 7/√5 B) 7/√13 C) √5 D) √13
48) let the base of a triangle lie along the line x= a and be of length 2a. the area of this triangle is a² if the vertex lie on the line.
A) x= -a B) x= 0 C) x= a/2 D) x= 2a
49) The equation of straight line passing through (1,2) and having intercept of length 3 between the straight lines 3x+ 4y= 24 and 3x+ 4y= 12 is
A) 7x+ 24y-55= 0
B) 24x+ 7y-38= 0
C) 24x - 7y-10= 0
D) 7x -24y+ 41= 0
50) The point (4,1) undergoes the following three transformations successively:
a) reflection about the line y=x
b) Translation through a distance 2 units along the positive direction of x axis
c) rotation through an angle π/4 about the origin in the clockwise direction.
the final position of the point is given by the coordinates
A) (1/√2, 7/√2)
B) (-2, 7√2)
C) (-1/√2, 7/√2) D) (√2, 7√2)
51) line L has intercepted a and b on the co-ordinate Axes. when the axes are rotated through a given angle, keeping the origin find, the same line has intercepts p and q. then
A) a²+ b²= p² + q²
B) 1/a²+ 1/b²= 1/p² + 1)q²
C) a²+ p²= b² + q²
D) 1/a²+ 1/p²= 1/b² + 1/q²
52) the point A(2,1) is translated parallel to the line x-y= 3 by a distance 4 units. if the new position A' is in third quadrant, then the coordinates of A' are
A) (2+2√2, 1+2√2)
B)(-2+√2, -1-2√2)
C) (2-2√2, 1-2√2) D) none
53) the point P(1,1) is translated parallel to 2x= y in the first quadrant through a unit distance. the coordinates of the new position of P are
A) (1±2/√5, 1±1√5)
B) (1±2/√6, 1±2√5)
C) (1/√5, 2/√5)
D) (2/√5,1/√5)
54) If a line joining two points A(2,0) and (3,1) is rotated about A in anticlockwise direction through an angle 15°, then the equation of the line in the new position is
A) √3x - y= 2√3
B) √3x + y= 2√3
C) x +√3 y= 2√3 D) none
55) An equilateral triangle has each side equal to a, if the coordinates of its vertices are (x₁, y₁); (x₂ ,y₂); (x₃, y₃), then the square of the determinants
x₁ y₁ 1
x₂ y₂ 1
x₃ y₃ 1 equals
A) 3a⁴ B) 3a⁴/4 C) 4a⁴ D) none
56) if a, b ,c are in AP, then ax+by+ c= 0 will always pass through a fixed point whose coordinates are
A) (1,-2) B) (-1,2) C) (1,2) D)(-1,-2)
57) If p and p' be perpendicular from the origin upon the straight lines x sect + y cosec t= a and x cos t - y sin t= a cos 2t respectively, then the value of the expression 4p²+ p'² is
A) a² B) 3a² C) 2a² D) 4a²
58) P(2,1), Q(4,-1), R(3,2) are the vertices of a triangle and if through P and R lines parallel to opposite sides are drawn to intersect in S, then the area of PQRS is.
A) 6 B) 4 C) 8 D) 12
59) If a line joining two points (2,0) and (3,1) is rotated about A in anticlockwise direction through an angle 15° such that the point B goes to C in the new position, then the coordinates of C are
A) (2+ 1/√2, √3)
B) (2+ 1/√2, √(3/2))
C) (2+ 1/√2, √3/2)
D) (2- 1/√2, -√(3/2))
60) If u= a₁x+b₁y+ c₁= 0 and
v= a₂ x+b₂y+ c₂= 0 and a₁/a₂ = b₁/b₂ = c₁/c₂, then u + kV= 0 represents
A) u= 0
B) a family of concurrent lines
C) a family of parallel lines D) n
61) If u= a₁x+b₁y+ c₁= 0 and
v= a₂ x+b₂y+ c₂= 0 and a₁/a₂ = b₁/b₂ ≠ c₁/c₂, then u + kV= 0 represents
A) a family of current lines
B) a family of parallel line
C) u= 0 or v= 0 D) none
62) If a straight line L is perpendicular to the line 5x- y=1 such that the area of the ∆ formed by the line L and the co-ordinate Axes is 5, then the equation of the line L is
A) x + 5y+5= 0
B) x + 5y±√2= 0
C) x + 5y±√5= 0
D) x + 5y± 5√2= 0
63) Let O be the origin and let A(1,0), B(0,1) be two points, if P(x,y) is a point such that xy> 0 and x+ y < 1, then
A) P lies other inside ∆ OAB or in third quadrant
B) P cannot be inside ∆ OAB
C) P lies inside the ∆ OAB. D) N
64) two sides of an isosceles triangle are given by the equation 7x - y+4= 0 and x + y -3= 0. If its third side passes through the point (1,-10), then the equations are
A) x -3y -7= 0 or 3x + y-31= 0
B) x -3y -31= 0 or 3x + y- 7= 0
C) x -3y -31= 0 or 3x + y +7= 0 D) n
65) a line is drawn from P(a,b) in the direction € with the x-axis, to meet Ax + By + C= 0 at Q. Then, the length PQ is equals to
A) |(Aa + Bb + C)/√(A²+ B²)|
B) - (Aa + Bb + C)/(A cos €+ B sin€)
C) (Aa + Bb + C)/(A cos €. B sin€)
D) -(Aa + Bb + C)/(A sin €+ B cos €)
66) point (1,2) and (-2,1) are
A) on the same said the line 4x+ 2y= 1
B) on the line 4x+ 2y= 1
C) on the opposite side of 4x+ 2y= 1 D) none
67) the equation of the lines on with the perpendiculars from the origin make 30° angle with x-axis and which form a triangle of area 50/√3 with the axes, are
A) x+ √3y±10= 0
B) √3x+ y ±10 = 0
C) x ± √3y-10= 0 D) none
68) The area of the triangle formed by y-axis, the straight line L passing through (1,1) and (2,0) and the straight line perpendicular to the line L and passing through (1/2,0) is
A)25/8 B)25/4 C)25/16 D)25/2
69) points on the line x+ y= 4 that lie at a unit distance from the line 4x+ 3y= 10 are
A)(3,1),(-7,11 B) (-3,7),(2,2)
C) (-3,7),(-7,11) D) none
70) The equation of the line passing through the intersection of the lines x - 3y= -1 and 2x+ 5y= 9 and at distance √5 from the origin is
A) 2x - y= 5 B) x+ 2y= 5
C) 2x+ y=5 D) x+ 2y= 1
71) the number of the line that are parallel to 2x+ 6y= -7 and have intercept of length 10 between the co-ordinate Axes is
A) 1 B)2 C)4 D) infinitely many
72) the line x+ y= 4 divides the line joining (-1,1) and (5,7) in the ratio k: 1, then the value of k
A) 1 B) 1/2 C) 3 D) none
73) the midpoint of the sides of a Triangles are (5,0),(5,12) and (0,12) the orthocentre of this triangle is
A)(0,0) B)(10,0) C) (0,24) D)(13/3,8)
74) the image of the point (1,3) in the line x+ y= 6 is
A)(3,5) B) (5,3) C) (1,-3) D) (-1,3)
75) a triangle ABC, right angled at A, has points A and B as (2,3) and (0,-1) respectively. If BC= 5units, then point C is
A)(-4,2) B) (4,2) C) (3,-3) D) (0,-4)
76) the equation of the line passing through the intersection of x -√3y+√3= 1 and x+ y= 2 and making an angle of 15° with the first line is
A) x- y= 0 B) x- y= -1
C) y= 1 D) √3x- y+1- √3= 0
77) in a rhombus ABCD the diagonals AC and BD intersect at the point (3,4). If the point A is (1,2) the diagonal BD has the equation
A) x- y-1= 0. B) x + y-1= 0.
C) x- y + 1= 0. D) x + y- 7= 0.
78) The algebraic sum of the perpendicular distances from A(x₁, y₁), B(x₂,y₂) and C(x₃,y₃) to a variable line is zero, then the line passes through
A) the orthocentre of ∆ABC
B) the centroid of ∆ ABC
C) the circumcentre of ∆ ABC
D) n
79) the ratio in which the line 3x- 2y +5= 0. divides the join of (6,-7) and (-2,3) is
A) 1:1 B) 7:37 C) 37:7 D) none
80) One vertex of equilateral triangle with centroid at the origin and one side as x+ y-2= 0 is
A)(-1,-1) B) (2,2) C) (-2,-2) D) n
81) the distance of the line x+ y- 8= 0. from (4,1) measured along the direction whose slope is -2
A)3√5 B) 6√5 C) 2√5 D) none
82) The area enclosed within the curve |x| + |y| = 1, is
A) 1 B) 2 C) 3 D) 4
83) the orthocentre of the triangle formed by the lines x+ y-1= 0, 2x+ 3y-6= 0 and 4x- y+4= 0 lies in
A) I quadrant B) II quadrant
C) III quadrant D) IV quadrant
84) If each of the points (a,4) and (-2, b) lies on the line joining the point (2,-1) and (5,-3), then the point P(a,b) lies on the line
A) x- 3y= 0 B) x +3 y= 0.
C) 2x- y +1= 0. D) 2x + 6y +1= 0.
85) The area bounded by the straight lines y=1 ±2x+ y-2= 0 is
A)1/2 B) 1 C) 3/2 D) 2
86) the locus of a point P which divides the line joining (1,0) and (2 cos€, 2 sin€) internally in the ratio 2:3 for all €, is a
A) straight line B) circle
C) pair of straight lines
D) parabola
87) A(-5,0) and B(3,0) are two vertices of a triangle ABC. its area is 20 square cms. the vertex C lie on the line x- y-2= 0. the co-ordinate of C are
A) (-7,-5) or (3,5)
B) (-5,-7) or (-3,-5)
C) (7,5) or (3,5)
D) (7,5),(-3,-5)
88) If the axes are rotated through an angle of 30° in the clockwise direction, the point (4, -2√3) in the new system was formally
A) (2,√3) B)(√3,-5) C)(√3,2) D)(2,3)
89) the ratio in which the line 3x + 4y +2= 0. divide the distance between 3x + 4y +5= 0, 3x + 4y -5= 0 is..
A) 7:3 B) 3:7 C) 2:3 D) none
90) If the extremities of the base of an isosceles triangle are the point (2a,0) and (0,a) and the equation of one of the sides is x.
A)5 B) 5/2 C) 25/2 D) none
91) the vertices of a ∆ OBC (0,0),B(-3,-1), C(-1,-3). The equation of a line parallel to BC and intersecting line OB and OC whose distance from the origin is 1/2, is
A) x + y +1/2= 0. B)x + y +1/2= 0.
C) x + y -1/√2 = 0.
D)2x + 6y +1/√2= 0.
92) the ratio in which the segment of the line joining (a,b) and (c,d) is cut by the line Ax + By +C= 0 is
A) (Aa+ Bb +C)/(Ac + Bd +C)
B)-(Aa+ Bb +C)/(Ac + Bd +C)
C) (Ac+ Bd +C)/(Aa + Bb +C)
D) -(Ac+ Bd +C)/(Aa + Bb+C)
93) The equation/s of the bisector/s of that angle between the lines x + 2y -11= 0, 3x -6y -5= 0 which contains the point (1,-3) is
A) 3x -19= 0 B) 3y -7= 0
C) 3x -11= 0 or 3y -7= 0 D) n
94) the lines 3x + 2y -24= 0 meets y axis at A and x axis at B. The perpendicular bisector of AB meets the line through (0,-1) parallel to x-axis at C. The area of the triangle ABC is
A)182 B) 91 C) 48 D) n
95) A ray of light coming from the point (1,2) is reflected at a point A on the x-axis and then passes through the point (5,3). the coordinates of the point A are
A)(13/5,0) B) (5/13,0) C) (-7,0) D) n
96) the area of the figure formed by the lines ax ± by ± c= 0 is
A) c²/ab B) 2c²/ab C) c²/2ab D) n
97) The incentre of the Triangle formed by the line x = 0, y = 0 and 3x + 4y -12= 0 is at
A)(1/2,1/2) B) (1,1) C) (1,1/2) D)(1/2,1)
98) If one vertex of an equilateral triangle is at (2,-1) and the base is x + y -2= 0, then the length of each side is
A)√(3/2) B) √(2/3) C)2/3 D)3/2
99) the area of the parallelogram formed by the lines 3x -4y+1= 0, 3x - 4y +3= 0, 4x 3 y - 1= 0, and 4x -3y -2= 0, is
A)1/6 B) 2/7 C) 3/8 D) none
100) points (1,3) and (5,1) are opposite vertices of a rectangle ABCD. if the slope of BD is 2, then its equation is
A) 2x - y -4= 0 B) 2x + y -4= 0
C) 2x + y - 7= 0 D) 2x + y +7 = 0
101) The line x + 2y -4 = 0 is translated parallel to itself by 3 units in the sense of increasing x and then rotated by 30° in the anticlockwise direction about the point where the sifted line cut the x-axis. The equation of the line in the new position is the line is it was an age 42 the line is
A) y= tan(€-30°)(x -4-3√5)
B) y= tan(30-€)(x -4-3√5)
C) y= tan(€+30°)(x +4+3√5)
D) y= tan(€-30°)(x +4+3√5)
102) The line PQ whose equation is x -y -2 = 0 cuts the x-axis at P and Q is (4,2). The line PQ is rotated about P through 45 in the anticlockwise direction. The equation of the line PQ in the new position is
A) y= - √2 B) y=2 C)x=2 D) x=-2
103) if the points (1,3) and (5,1) are two opposite vertices of a rectangle and the other two vertices lie on the line 2x -y +c, then the value of c is
A)4 B) -4 C) 2 D) none
104) In the above question the coordinates of the other two vertices are
A)(2,0),(4,-4) B) (2,4),(4,0)
C)(-2,0),(4,-4). D) (2,0)(-4,4)
105) If the vertices of a diagonal of a square are (2,4) and (-2,2), then its other two vertices are at
A)(1,-1),(5,1) B) (1,1),(5,-1)
C)(1,1),(-5,1). D) none
106) the equations of two sides of a square whose area is 25 square units are 3x - 4y = 0 and 4x +3y = 0. The equation of the other two sides of the square are
A) 3x - 4y ±25 = 0, 4x +3y ±25 = 0
B) 3x - 4y ±5 = 0, 4x +3y ±5 = 0
C) 3x - 4y ±5 = 0, 4x +3y ±25 =0 D) none
107) The equations of the lines through (-1,-1) and making angle 45° with the line x - 4y = 0, are given by
A) x²- xy+ x -y= 0
B) -y²+xy+ x -y= 0
C) xy+ x +y= 0
D) xy+ x +y+1= 0
108) if the lines ax+2y+1=0, bx +3y+1= 0, cx+4y+ 1= 0 are concurrent, then a, b, c are in
A) AP B) GP C) HP D) none
109) two vertices of a triangle are (5,-1), and (-2,3). If the orthocentre of the triangle is the origin, then coordinates of the third vertex are
A)(4,7) B) (-4,-7) C) (-4,7) D) n
110) If the foot of the perpendicular from the origin to a straight line is at point (3,-4). then the equation of the line is
A) 3x - 4y= 25 B) 3x - 4y= -25
C) 4x + 3y= 25 D) 4x - 3y= -25
111) a rectangle has two opposite vertices at the point (1,2) and (5,5). If the other vertices lie on the line x= 3, then their coordinates are
A) (3,1),(3,3) B) (3,1),(3,6)
C) (3,1)(3,4) D) none
112) the orthocentre of the Triangle formed by the line xy= 0 and x +y= 1 is
A)(1/2,1/2). B) (1/3,1/3)
C) (0,0). D) (1/4,1/4)
113) If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
A) square B) circle
C) straight line
D) two intersecting lines.
114) A line passes through (2,2) and is perpendicular to the line 3x +y= 3. Its y-intercept is
A)1/3 B) 2/3 C) 1 D) 4/3
115) the orthocentre of the triangle with vertices (2, (√3 -1)/2),(1/2,-1/2) and (2,-1/2) is
A) (3/2, (√3 -3)/6)
B) (2,-1/2) C) (5/4, (√3 -3)/4)
D) (1/2,-1/2)
116) The equation to a pair of opposite sides of a parallelogram are x² -5x+6= 0 and y² -6y+5= 0. the equation to its diagonal are
A) x +4y= 13 and 4x -y= 7
B) 4x +y= 13 and 4x -4y= 7
C) 4x +y= 13 and 4x -y= 7
D) y +4x= 13 and y+4x= 7
117) The distance between the parallel lines 2x -y= -4 and 6x -3y= -5 is
A)17/√3 B) 1 C) 3/√5 D) 17√5/15
118) P is a point on either of the two lines y - √|x|= 2 at a distance of 5 units from their point of intersection. the coordinates of the foot of the perpendicular from P on the bisector of the angle between them are
A) (0, (4+5√3)/2) or (0, (4-5√3)/2)
depending on which the point P is taken.
B) (0, (4+5√3)/2) C) (0, (4-5√3)/2
D) (5/2,5√3/2)
119) If one of the diagonals of a square is along the line x - 2y=and one of its vertices is (3,0), then its sides through this vertex are given by the equations
A) y- 3x = -9 or 3y+x-3=0
B) y+ 3x+9 =0 or 3y+x-3=0
C) y- 3x = -9 or 3y-x+3=0
D) y- 3x = -3 or 3y+x+9=0
120) The line which is parallel to x-axis and crosses the curve y=√x at an angle of 45° is
A) x=1/4 B) y=1/4 C) y= 1/2 D) y=1
121) If the two pairs of lines x²- 2mxy - y²= 0 and x²- 2nxy - y²= 0 are such that one of them represents the bisector of the angles between the other, then
A) mn+1= 0 B) mn-1= 0
C) 1/m + 1/n= 0 D) 1/m - 1/n= 0
122) P(3,1), Q(6,5) and R(x,y) are three points such that the angle PQR is a right angle and the area of ∆ RQP= 7, then the number of such points R is
A) 0 B) 1 C) 2 D) 4
123) The equation of the straight line which passes through the point (1,-2) and cuts off equal intercepts from the axes will be
A) x+y= 1 B) x- y= 1 c) x+y= -1 d) x-y= -2
124) the equation of the straight line which is perpendicular to y= x and passes through (3,2) will be given by
A)x-y= 5 B)x+y= 6 C)x+y= 1 D) x-y= 1
125) The distance between the lines 3x+ 4y= 9 and 6x+ 8y= 15 is
A) 3/2 B) 3/10 C) 6 D) none
126) The equation of the straight line passing through (1,2) and perpendicular to x+y= -1 is
A) x-y= 1 B) x+y= -1
C) x -y= 2 D) y -x -2= 0
127) The straight lines x+y= 0, 3x+y= 4, x+ 3y= 4 form a triangle which is
A) isosceles B) equilateral
C) right angled D) none
128) given four lines with equations x+2y= 3, 3x+4y= 7, 2x+ 3y= 4, 4x+ 5y= 6, then they are
A) all concurrent
B) the sides of quadrilateral
C) none
129) If a line is perpendicular to the line 5x-y= 0 and forms a triangle with co-ordinate Axes of area 5 sq.units, then its equation is..
A)x+5y ±5√2= 0
B) x -5y±5√2= 0
C) 5x+y± 5√2= 0
D) 5x - y±5√2= 0
130) Orthocentre of the triangle whose sides are given by 4x -7y= - 10, x+y= 5 and 7x+ 4y= 15 is
A) (-1,-2) B)(1,-2) C)(-1,2) D)(1,2)
131) the line (p+2q)x +(p-3q)y= p - q for different values of P and Q passes through the fixed point.
A)(3/2,5/2) B) (2/5,2/5)
C) (3/5,3/5) D) (2/5,3/5)
132) the distance between the lines 4x+3y=11 and 8x+6y=15 is
A) 7/24 B) 4 C) 7/10 D) none
133) the point (4,1) undergoes the following two successive transformations:
I) reflection About line y= x
ii) rotation through a distance 2 units along the positive x axis.
then the final coordinates of the point are
A)(4,3) B) (3,4) C) (1,4). D) 7/2,7/2)
134) The straight lines x+y=4, 3x+y=4, x+3y=4 form a triangle which is
A) isosceles B) right angled
C) equilateral D) none
135) The orthocentre of the triangle formed by (0,0),(8,0),(4,6) is
A)(4,8/3) B)(3,4) C) (4,3) D)
(-3,4)
136) A point equidistant from the line 4x+3y=-10, 5x -12y=26, and 7x+24y=50 is
A)(1,-1) B) (1,1) C) (0,0) D) (0,1)
137) the lines 2x+y=1, ax+3y=3 and 3x+2y=2 are concurrent for
A) all a B) a=4 only
C) -1≤ a≤ 3 D) a> 0 only
138) The diagonal of the parallelogram whose sides are lx+my+n=0, lx+my+ n'=0, mx+ ly + n=0, mx+ly+ n'=0 included an angle
A)π/3 B) π/2
C) tan⁻{(l²⁻m²)/(l²+m²)}
D)tan⁻{(2lm)/(l²+m²)}
139) the equation to the sides of a triangle are x- 3y=0, 4x+ 3y=5, and 3x+y=0. the line 3x - 4y=0 passes through
A) Incentre B) the centroid
C) the circumcentre
D) the orthocentre of the triangle
140) A straight line through P(1,2) is such that its intercept between the axes is bisected at P. Its equation is
A) x+ 2y=5 B) x - y= -1
C) 2x+y=3 D) 2x+y=4
141) area of the quadrilateral formed by the lines |x|+|y|=1 is
A) 4 B) 2 C) 8 D) none
142) If the line y=mx meets the lines x+ 2y=1 and 2x- y=3 at the same point, the m is equal to
A) 1 B) -1 C) 2 D) -2
EXERCISE - 3
1) The set of lines ax+ by+ c= 0 where 3a+ 2b+ 4c= 0 is concurrent at the point____
2) the orthocentre of the Triangle formed by the line x+ y+ =1, 2x+ 3y= 6 and 4x - y+ 9= 0 lies in quadrant number___
3) the area enclosed by the lines |x|+ |y| = 2 is _____
4) if the image of a point (4,-6) by a line mirror be (2,1) then the equation of the line mirror is__
5) If the points (-2,-5(,(2,-2),(8,a) are collinear, then the value of a is ____
6) If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is _____
7) if a and b are real numbers between 0 and 1 such that the points (a,1),(1,b) and (0,0) form an equilateral triangle, then s = ____, b= ______.
8) let the algebraic sum of the perpendicular distances from the point (2,0),(0,2),(1,1) to a variable line zero, then the line passes through a fixed point whose coordinates are _____.
9) If A and B are points in the plane such that PA/PB= k, (a constant) for all P on a given circle, then the value of k cannot be equal to _____.
10) the equation of a line through the point (1,2) whose distance from the point (3,1) has the greatest possible value is___.
11) a square has two opposite vertices at the points (3,4) and (1,-1), then the other vertices are ___.
12) If the lines y= 3x+1 and 2y= x+3 are equally inclined to the line y= mx+4, then the value of m is ____
13) A line meets OX, OY in the points A, B respectively. If the line AB always passes through a fixed point (h,k), then the locus of the midpoint of AB has equation__
14) The equation of the locus of the midpoints of the chords of the circle 4x²+ 4y² -12x+4y+1=0 that subtend an angle of 2π/3 at its centre is_____
15) The orthocentre of the triangle formed by the lines x= 0, my= m²x+a and ly= t²x+a is the point (_____, _______).
16) if x cos€ + y si €= - sin€ tan€ be a straight line, then the length of the perpendiculars on this line from the points (m²,2m(,(mn, m+n) and (n²,2n) form a ____
17) Three distinct lines tᵢx+y= 2atᵢ ; i= 1,2,3 are concurrent if ----. (a≠0).
18) If the ends of a rod of length l move on two mutually perpendicular lines, then the locus of the point on the rod which divides it in the ratio 1:2 is ___.
19) if the vertices of a triangle have integral coordinates, then the triangle cannot be____
20) the area of the triangle formed by the line y= mx+ c, y= nx + d and x= 0 is_____
21) If the extremities of the diagonal of a square are (1,1),(-2,-1), then the equation of the Other diagonal is____
22) if the real line ax+by+c= 0, bx+cy+a= 0 and cx+ay+b= 0 are concurrent, then _____.
23) if the sides of a square lie on the lines 5x -12y - 65= 0, and 5x -12y+26 = 0, then its area is ___
24) the equations of the sides of a square whose Centre is at the origin and a vertex at (1,2) are__
25) The equation of one side of a rectangle is 3x -4y-10= 0 and the coordinates of two of its vertices are (-2,1) and (2,4). then the area of the rectangle is____
26) The points (1,3) and (5,1) are two opposite vertices of a rectangle and the other two vertices lie on the line 2x - y+c= 0, then the value of c is ___
27) two consecutive sides of a parallelogram are 4x+5y= 0 and 7x+2y= 0, if the equation to one of its diagonal is___
28) If the quadrilateral whose sides are given by aᵣx + bᵣy + cᵣ = 0, r= 1,2,3,4 be concyclic, then___
29) The equations of two sides of a square are 3x+4y - 5= 0 and 3x+4y -15 = 0, the equation of third side which has a point (6,5) on it is_____
30) A line is such that its segment between the lines 5x - y +4 = 0 and 3x+ 4y -4 = 0 is bisected at (1,5). its equation is__
31) If t₁ + t₂ + t₃ = -t₁t₂t₃, then the orthocentre of the triangle formed by the points (at₁t₂ , a(t₁ + t₂)), (at₂t₃, a(t₂ + t₃))) and (at₃t₁ , a(t₃ + t₁)) lies on____
32) If the transversal y= mᵣx; r = 1,2,3 cut off equal intercepts on the transversal x+ y=1, then 1+ m₁, 1+m₂, 1+m₃ are in____
33) the equations of two families of lines are x+y-3+ K(2x -y)= 0 and 3x - y + M(x - 2y+1)= 0. the equation of the line which belongs to both the families is__
34) If al+ bm + cn= 0 then the family of lines lx+ my + n= 0 are concurrent at____
35) A straight line is such that the algebraic sum of the perpendicular drawn upon it from any number of fixed points is zero. then the line always passes through______
36) the determinantal equation
x y 1
x₁ y₁ 1 = 0
x₂ y₂ 1
represents geometrical a ____ passing through ____ and _____
37) If the straight lines ax+ by + p= 0 and x cos€ + y sun€ - p= 0 encloses an angle π/4 between them, and meet the straight line x sin € - y cos€ = 0 in the same point, then the value of a²+ b² is equals to _____
38) If a, b, c are in AP, then the family of lines ax+ by + c= 0 pass through a fixed point whose coordinates are_____.
EXERCISE -4
Write the truth value T/F of the following statements:
1) x₁ y₁ 1 a₁ b₁ 1
x₂ y₂ 1 = a₂ b₂ 1
x₃ y₃ 1 a₃ b₃ 1
Then the two triangles with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) and (a₁,b₁),(a₂,b₂),(a₃,b₃) must be congruent.
2) the coordinates of the vertices of an equilateral triangle are integers.
3) the straight line 5x+ 4y= 0 passes through the point of intersection of the straight lines x+ 2y -10= 0 and 2x+ y +5= 0.
4) the lines 2x+ 3y +19= 0 and 9x+ 6y -17= 0 cut the co-ordinate Axes in concyclic points.
5) no tangent can be drawn from the point (5/2,1) to the circumcircle of the triangle with vertices (1,√3),(1,-√3), (3,-√3).
6) the point P(-3,2) lies inside the triangle whose sides are given by the equations x+ y -4= 0, 4x-y -31= 0, 3x -7y +8= 0
7) The points (2,-5) and (-1,4) are equidistant from the line 3x+ y +5= 0
8) the family of line ax+ by +c= 0 where 3a+ 2b +4c= 0 pass through the fixed points (-3,-2).
9) the origin lies inside the triangle whose vertices are given by the equations 7x - 5 y -11= 0, 8x+3y +31= 0, x+ 8y -19= 0.
10) only one straight line can be drawn through the origin at equal distances from the points A(2,2) and B(4,0).
11) the line x(a+2b) + y(a-3b)= a - b passes through a fixed point for different values of a and b.
12) the product of the perpendiculars from the points (√5,0),(-√5,0) to the straight line 2x cos€ - 3y sin€ = 6 is independent of €.
13) The lines ax+ (b+c)y = p, bx+(c+a) y and cx+ (a+b)y = p encloses an equilateral triangle.
14) If t₁, t₁ are distinct and different from zero, then the line joining the points (t₁², t₁) (t₂², t₂) passes through the origin.
15) point Q is symmetric to (4,-1) with respect to the bisector of the first quadrant. Then the length of PQ is 5√2.
16) Two vertices of a triangle are (-4,3) and B(4,-1) and orthocentre is the point O'(3,3). Then the third vertex C is the point (4,5).
17) y= 10ˣ is the reflection of y= logx in the straight line y= x.
18) the lines ax + by +c= 0, bx +c y +a= 0,and cx +a y +b= 0 are concurrent if a³+ b³+ c³= 3abc.
19) If 2p is the length of the perpendicular from the origin to the line x/a + y/b = 1, then a², 8p², b² are in HP.
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