DISTANCE FORMULA
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EXERCISE-1
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1) Find the distance between the points:
a) A(2,3) , B(-6,3). 8
b) C(-1,-1), D(8,11). 15
c) P(-8, -3), Q(-2, -5). 2√10
d) R(a+b, a- b), S(a-b), a+b). 2b√2
e) (ap², 2ap) and (aq², 2aq).
f) (a + p sinx, b + p cosx) and
(a + q sinx, b+ q cosx). |p- q|
g) (am², 2am), (a/m², -2a/m). a(m + 1/m)²
h) Prove that the distance between the points (a cos t, a sin t) and (cos p, a sin p) is |2a sin(k - p)/2|.
i) Find the length of the sides of the triangle whose vertices are A(3,4), B(2, -1) and C(4, -6). √26,√29,√101
j) A line is of length 10, and one end is at the point (-3,2). If the ordinate of the other end be 10, Prove that the abscissa will be 3 or -9.
k) Which one among the points (2,3), (-3,1) and (0,4) is nearest to the origin ? (-3,1)
l) Find the distance between the points A(x₁,y₁) and (x₂, y₂), when
i) AB is parallel to the x-axis.
ii) AB is parallel to the y-axis.
m) A is a point on the x-axis with abscissa -8 and B is a point on the y-axis with the ordinate 15. Find the distance AB. 17 units
2) Find the distance of the following point from the origin:
a) (6, -6). 6√2
b) (-7, -24). 25
c) (a+b, a - b). √{2(a²+b²)}
d) (ap+bq), aq - bp). √{(a²+b²)(p²+ q²)
e) Show that the distance of the point (a cosx + b sinx, a sinx - b cosx) from origin is √(a²+b²) units
3) a) If the distance between the points (-3,3) and (4, y) be 5√2 units, find the value of y. 4 or 2
b) If the distance between the points (3,5) and (x, 8) be 5 units, find the value of x. 7 or -1
c) A point with ordinate 3 lies on the line joining two points (4,1) and (-2,7). Find its abscissa. 2
d) If the distance between the points P(3,4) and Q(5, k) be √13 units, find the coordinates of Q. (5,7) or (5, 1)
e) Find the radius of the circle that has its centre at (0, -4) and passes through (√13,2). 7
4)a) If a point P(x,y) is equidistance from the point A(6, -1) and B(2,3). Find the relation between x, y. x- y= 3
b) If the points (a,b) and (b, a) are equidistant from the point (x,y), prove that x= y.
c) Find the coordinates of the point which lies on y- axis and is equidistant from the point (2,3) and (-1,2). (0,4)
d) Find a point on the x-axis which is equidistant from the point A(7,6) and (-3, 4). (3,0)
e) For what value of k will the point (cos 2k, sin 2k) be equidistant from the two axes ? nπ/2 ± π/8, where n is zero or any integer.
f) Find a point on the y-axis which is equidistant from A(-4,3), B(5,2). (0,-2).
g) Find the coordinates of the point equidistant from (2,6), (-2,2) and (-5, -1). (-2,3)
EXERCISE--2
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1) Show that the points are the vertices of an isosceles triangle.
a) (1,4), (4,1), (8,8).
b) (-1,5), (3,2), (-1, -1).
c) (3,3), (-3, -5), (-5,-3).
2) Show that the vertices are right angled triangle.
a) (3,4), (-1,7) , (-3, -4).
b) (1,3), (6,5), (5, -7).
c) (-1,3), (0,5), (3,1).
3) Show that the points are the vertices of an isosceles right angled triangle.
a) A(7,10), B(-2,5) and C(3, -4).
b) (3,1), (9,7), (-3,7).
c) (3,3), (-2, -2), (8, -2).
4) Show that the points are the equilateral triangle
a) A(1,1), B(-1, -1) and C(-√3, √3).
b) (3,2), (1,0), (2- √3, 1+ √3).
c) If (1,1), (-1, -1), (a, -a) are the vertices then the value of a. ±√3
d) In an equilateral triangle ABC, if the coordinates of the vertices B and C are (2a, 6a) and (2a+ √3 a, 5a) respectively, find the coordinates of the vertex A. (2a, 4a) or (2a+ √3 a, 7a)
e) If the origin is situated inside an equilateral triangle and the coordinates of two vertices of the triangle be (3,2) and (-3,2), then find the coordinates of the third vertex. (0, 2- 3√3)
f) Vertices of an equilateral triangle are (a,b), (a+ r cos t, b + r sin t), (a + r cos k, b + r sin k), find the value of | t - k|. π/3
5) Show that the points are the angular parts of rectangle.
a) A(2,-2), B(8,4), C(5,7) and D(-1,1).
6) Show that are the vertices of a square.
a) A(3,2), B(0, 5), C(-3,2) and D(0,-1).
b) (-2,-7), (2,-4), (-1,0), (-5,-3).
c) (2,1), (0,0), (-1,2), (1,3).
7) Show that are the vertices of a parallelogram.
a) A(1,-2), B(3, 6), C(5,10) and D(3, 2)
b) (-2,-1), (1,0), (4,3) and (1,2).
c) (-1,2), (6,-3), (4,-10), (-3,-5).
d) (2,4), (3,8), (5,1), (4,-3).
8) Show that the points are the vertices of a rhombus.
a) A(2, -1), B(3,4), C(-2,3) and D(-3, -2)
b) (0,5), (-2,-2),(5,0),(7,7)
c) (0,0), (0,10), (8,16), (8,6).
d) (2,5), (6,8), (9,12), (5,9).
EXERCISE --3
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** Find the area of:
1) triangle:
a) A(-3, -5), B(5,2) and C(-9, -3). 29
b) (4,4),(3,-16),(3,-2)
c) (a,0), (0,b), (x,y). 1/2(bx+ ay- ab)
d) (a, 1/a), (b, 1/b), (c, 1/c).
e) (a, b+c), (a, b - c), (-a, c). |2ac|
f) (0,0), (cos t, sin t), (cos 2t, sin 2t). 1/2 |sin t|
g) (a, bc), (b, ca), (c, ab)
h) Sides are y+2x= 3, 4y+ x=5, 5y+3x= 1. 3.5
i) (a cos t, b sin t), (a cos k, b sin k) and (a cos g, b sin g).
j) A(3,4), B(5,2), C(x,y); if the area of the triangle ABC is 3 square units, show that x+ y -10 = 0.
k) The coordinates of the points A, B, C are (6,3), (-3,5), (4,-2) respectively and that of the point P(x,y). Show that the ratio of the areas of ∆ PBC and ∆ABC is |x+ y -2|/7.
l) A and B are two points (3,4) and (5,-2). Find point P such that PA= PB and ∆PAB= 10. (7,2)
2) Find the area of quadrilateral whose vertices are
a) A(-4, 5), B(0,7), C(5, -5) and D(-4, -2). 60.5
b) (-2,-3), (6,-5), (18,9), (0,12). 206
c) (1,1), (3,4), (5,-2), (4,-7). 41/2
3) If the area of the quadrilateral whose angular points A,B, C, D taken in order are (1,2), (-5,6), (7,-4), (-2,k) be zero, find k. 3
4) The coordinates of quadrilateral ABCD are (-3,4),(-1,-2),(5,6) and (x, -4) respectively. If ∆ABD = 2∆ACD, find the value of x. 14.2
5) Show that the points are collinear:
a) A(-5,1), B(5, 4) and C(10, 7) are collinear.
b) A(3, -2), B(5, 2) and C(8,8) are collinear.
c) (1,2), B(2,-1), (3,-4).
d) (a, b+c), (b, c+a), (c, a+b).
6) If the three points (1,2), (2,4),(t,6) are collinear, find t. 3
7) Find the value of k for which the points A(-2,3), B(1,2) and C(k, 0) are collinear. 7
8) Find the area of the triangle with Vertices A(3,1), B(2k, 3k), C(k, 2k). Show that the three distinct points A, B, C are collinear when k= - 2.
9) If (a,0), (0,b) and (1,1) are collinear show that 1/a + 1/b = 1.
10) Find the condition of collinearity of the points (a,b), (A' , b') and (a- A' , b- b').
11) The three points P, Q, R are collinear. If the coordinates of the points P and Q are (3,4), (7,7) respectively and PR= 10 units, find the coordinates of the point R. (11, 10) or (-5,-2)
12) If the points (a,b),(A' , b') and (a- a', b - b') are collinear, show that ab'= a'b.
EXERCISE-4
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1) Find the coordinates of the midpoint of the line joining points
a) (-2,-5), (3,-1). (1/2,-3)
b) (0,0),(8,-5). (4,-5/2)
c) (-4,3),(6,-7). (1,-2)
d) Find the midpoints of the sides of a triangle whose vertices are A(1,-1), B(4,-1), C(4,3). (5/2,-1), (4,1),(5/2,1)
e) Find the centre if the end points of a diameter are A(-5,7) and B(3,-11). (-1,-2)
f) If M is the midpoint of AB, find the coordinates of:
i) A if the coordinates of M and B are M(2,8) and B(-4,19). (8,-3)
ii) B if the coordinates of A and M are A(-1,2), M(-2,4). (-3,6)
g) The vertices of ∆ ABC are A(-1,3), B(1,1) and C(5,1). Find the length of the median to
i) AB. √26
ii) AC. √2
iii) BC. 2√5
h) A circle has its centre at the origin and a radius of √12. State whether each of the following points is on, outside or inside the circle:
i) (1, -√7). Outside
ii) (3,5). Outside
iii) (2, 2√2). On
i) Prove that the midpoint of the line segment joining the points (2,1) and (6,5) lies on the line joining the points (-4,-5),(9,8).
j) If R(8,17) be the midpoint of the line segment joining the points P(-5,-3) and Q(x,y); find the coordinates of Q. (21,37)
2) Find the coordinates of the point which divides the join of:
a) A(-5, 11) and B(4, -7) in the ratio 2 :7. (-3,7)
b) (5,-2), (9,6) in the ratio of 3:1. (8,4)
c) (-4,4),(1,7) in 2:1 externally. (6,10)
d) (3,4),(-6,2) in 3:2 internally. (21,8)
e) (a+b, a-b) and (a-b, a+b) internally and externally in the ratio a: b.
3) a) In what ratio is the line segment joining the points A(-4,2) and B(8, 3) divided by the y-axis ? Also find the point of intersection.
1:2, (0, 7/3)
b) Find the ratio in which the x-axis cuts the join of the points A(4,5) and B(-10, -2). Also, find the point of intersection. 5:2, (-6,0)
c) In what ratio does the point (1,-7/2) divides the join of (-2,-4) and (2,-10/3) ?. 3:1
d) In what ratio is the line joining the points:
i) (2,-3) and (5,6) divided by the x-axis. 1:2
ii) (3,-6) and (-6,8) divided by the y-axis ? 1:2
e) Find the ratio in which the axes divide the line joining the points (2,5) and (1,9). 5:9 and 2:1 externally
f) In what ratio is the line joining A(-1,1) and B(5,7) divided by the line x+y = 4. 1:2
g) Find the coordinates of the points of trisection of the line joining the points (2,3) and (6,5). (10/3,11/3),(14/3,13/3)
h) A line segment directed from (-3,2) to (1,-4) is trebled. Find the coordinates of the terminal point. (9,-16)
i) The line segment joining the points (2,-2) and (4,6) is extended each way a distance equal to half its own length; find the coordinates of its terminal points. (5,10),(1,-6)
j) The line joining the points (3,2) and (6,8) is divided into four equal parts, find the coordinates of the points of section. (15/4,7/2), (9/2,5),(21/4,13/2)
k) Determine the coordinates of the vertices of a triangle if the middle points of its sides have the coordinates (2,-3),(4,2) and (-5,-2). (-7,-7),(11,1),(-3,3)
l) If the point (9,2) divides the line segment joining the points P(6,8) and Q(x,y) in the ratio 3:7, find the coordinates of Q. (16,-12)
m) If the point (6,3) divides the segment of the line P(4,5) to Q(x,y) in the ratio 2:5, find the coordinates (x,y) of Q. What are the coordinates of the midpoint of PQ. (11,-2), (15/2,3/2)
4) Find the centroid of the triangle:
a) (-4, 6),(2,-2),(2,5). (0,3)
5) Find the coordinates of the in-centre of the triangle whose vertices are:
a) (-36,7),(20,7),(0,-8). (-1,0)
b) (-1,5),(3,2),(-1,-1). (1/2,3/4)
c) (0,0),(2,0),(0,3).
d) (0,0),(3,0),(0,4). (1,1)
6) Find the ortho-centre of the triangle with vertices
a) (2,3),(4,6),(-1,1). (14,-9)
b) (-3,1),(2,4),(5,1). (2,4)
c) The ortho-centre of a triangle whose vertices are (-3,1) and (2,4) is (2,4). Find the third vertex of the triangle. (5,-1)
7) Find the circum-centre of the triangle whose vertices are:
a) (-3,0),(3,2),(5,-2). (6/7,-11/7)
b) (-2,3),(1,2),(4,-1). (-3,-5)
c) The coordinates of the circum-centre of a triangle ABC are (8,3). If the coordinates of the vertices A, B ,C are (x, -9),(y, -2( and (-5,3) respectively, then find the values of x and y. x= 3,13 and y= -4,20
d) Find the circum-centre of the triangle formed by three points (-3,1),(1,3) and (3,0). Find the circum-radius of the triangle. (-1/16, 1/8); √(2405)/16 units
8) Find the area of ∆ ABC, the midpoints of whose sides AB, BC and CA are D(3, -1), E(5, 3) and F(1, -3) respectively. 8
9) If the points A(-2, -1), B(1,0), C(x,3) and D(1, y) are the vertices of a parallelogram, find the values of x and y. 4, 2
10) If the coordinates of the points A, B and S are respectively (at², 2at), (a/t², -2a/t) and (a,0), Prove 1/SA + 1/SB = 1/a.
EXERCISE--5
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1)
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