1) AB= √{(x₁ - x₂)² + (y₁ - y₂)²}
2) Internally: (m₁x₂ + m₂x₁)/(m₁ + m₂) and (m₁y₂ + m₂y₁)/(m₁ + m₂)
3) When externally: (m₁x₂ - m₂x₁)/(m₁ - m₂) andb(m₁y₂ + m₂y₁)/(m₁ - m₂).
4) Midpoint: (x₁ + x₂)/2 and (y₁ + y₂)/2
5) Centroid: (x₁ + x₂ + x₃)/3 and (y₁ + y₂ + y₃)/3
CHAPTER - 1
Find the distance between the following pairs of points.
1) (2,3) and (5,7)
2) (4,-7) and (-1,5).
3) (-3,-2) and (-6,7), the axes being inclined at 60°.
4) (a,a) and (a,b).
5) (b + c, c+ a) and (c + a, a+ b).
6) (a cosα, a sin α) and (a cosβ, a sinβ).
7) (am₁², 2am₁) and (am₂², 2am₂).
8) Lay down in a figure the positions of the points (1,-3) and (-2,1), and show that the distance between them is 5.
9) Find the value of x₁ if the distance between the points (x₁,2) and (3,4) be 8.
10) A line is of length 10 and one end is at the point (2,-3); if the abscissa of the other end be 10, show that its ordinate must be 3 or -9.
11) Show that the points (2a,4a), (2a, 6a), and (2a + √3a, 5a) and the vertices of an equilateral triangle whose side is 2a.
12) Show that the points (-2,-1),(1,0),(1,3),(1,2) are at the vertices of a parallelogram.
13) Show that the points (2,-2),(8,4),(5,7) and (-1,1) are at the angular of a rectangle.
14) Show that the points (-1/14, 30,14) is the centre of the circle circumscribing the triangle whose angular points are (1,1), (2,3), and (-2,2).
# Find the coordinates of the point which
15) divides the line joining the points (1,3) and (2,7) in the ratio 3:4.
16) divides the line joining the points (1,3) and (2,7) in the ratio 3:4.
17) divides, internally and externally, the line joining (-1, 2) to (4, -5) In the ratio 2:3.
18) divides, internally and externally, the lines joining (-3,-4) to (- 8,7) in the ratio 7:5.
19) The line joining the points (1,-2) and (-3,4) is trisected; find the coordinates of the points of trisection.
20) The line joining the points (-6,8) and (8,-6) is divided into 4 equal parts; find the co-ordinates of the points of section.
21) Find the coordinates of the points which divide, internally and externally, the line joining the point (a+ b, a - b) to the point (a - b, a+ b) in the ratio a: b.
22) The co-ordinates of the vertices of a triangle are (x₁, y₁), (x₂, y₂) and (x₃, y₃). The line joining the first two is divided in the ratio 1: k, and the line joining this point of division to the opposite angular point is then divided in the ratio m : k+1. Find the coordinates of the latter point of section.
23) Prove that the coordinates, x and y, of the middle point of the line joining the point (2,3) to the point (3,4) satisfy the equation x+ y +1=0.
24) if G be the concentroid of a triangle ABC and O be any other point, prove that
3(GA²+ GB²+ GC²)= BC²+ CA²+ AB²,
and OA²+ OB²+ OC⅖= GA²+ GB²+ GC²+ 3GO².
25) Prove that the line joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one another.
26) A, B, C, D... are points in a plane whose coordinates are (x₁, y₁), (x₂, y₂) and (x₃, y₃)... AB is bisected in the point G₁ : G₁C is divided at G₂ in the ratio 1:2; G₂D is divided at G₃ in the ratio 1:3; G₃E at G₄ in the ratio 1:4, and so on until all the points are exhausted. Show that the coordinates of the final so obtained are
(x₁ + x₂+ x₃+....xₙ)/n and (y₁+ y₂ + y₃+....yₙ)/n
(This point is called the Centre of Mean Position of the n given points).
27) Prove that a point can be found which is at the same distance from each of the four points.
(am₁, a/m₁), (am₂, a/m₂), (am₃, a/m₃), and (a/(m₁m₂m₃) , am₁m₂m₃).
1) 5 2) 13 3) 3√7 4) √(a²+ b²) 5) √(a²+ 2b²+ c²- 2ab - 2bc) 6) 2a sin{(α-β)/2} 7) a(m₁ - m₂) {√(m₁ + m₂)²+4}. 9) 3± 2√15 15) 16) (-2,-9) 17) 19) (-1/2,0); (-3/2,2)
Area of Triangle=
∆= 1/2(x₁y₂ - x₂y₁ + x₂y₂ - x₃y₂ + x₃y₁ - x₁y₃).
EXERCISE - B
Find the areas of the triangles the coordinates of whose angular points are respectively.
1) (1,3), (-7,6) and (5,-1).
2) (0,4),(3,6), and (-8,-2).
3) (5,2),(-9,-3) and (-3,-5).
4) (a, b + c), (a, b - c) and (-a, c).
5) (, c+ a), (a, c) and (-a, c - a).
6) (a cosθ₁, b sinθ₁), (a cosθ₂, b sinθ₂) and (a cosθ₃, b sinθ₃).
7) (am₁², 2am₁), (am₂², 2am₂) and (am₃², 2am₃).
8) {(am₁m, a(m₁+ m₂)}, {am₂m₃, a(m₂ + m₃)}, and {am₂m₁, a(m₃ + m₁)}.
9) (am₁, a/m₁), (am₂, a.m₂) and (am₂, a/m₃).
Prove (by showing that the area of the triangle formed by them is zero)
10) (1,4),(3,-2) and (-3,10).
11) (-1/2,3),(-5,6) and (-8,8).
12) (a, b + c), (b, c + a) and (c, a+ b).
Find the areas of the quadrilateral the coordinates of whose angular points, taken in order, are
13) (1,1),(3,4),(5,-2) and (4,-7)
14) (-1,6),(-3,-9),(5,-8) and (3,9).
15) If O be the origin, and if the coordinates of any two points P₁ and P₂ be respectively (x₁, y₁) and (x₂, y₂), prove that
OP₁. OP₂ cosP₁OP₂ = x₁x₂ + y₁y₂.
To find the length of the straight line joining two points whose coordinates are given:
r₁²+ r₂²- 2r₁r₂ cos(θ₁+ θ₂).
To find the area of a triangle the coordinates of whose angular points are given
∆ ABC= (1/2) [r₂r₃ sin(θ₃ - θ₂) + r₃r₁ sin(θ₁ - θ₃) + r₁r₂ sin(θ₂ - θ₁).
EXERCISE - C
Lay down the position of the points whose polar coordinates are
1) (3,45°).
2) (-2, -60°)
3) ( 4, 135°)
4) (2, 330°)
5) -1, 180°)
6) (1, 210°)
7) (5, 675°).
8) (a, π/2).
9) (2a, -π/2).
10) (-a, π/6).
11) (-2a, - 2π/3)
Find the lengths of the straight lines joining the pairs of points whose polar coordinates are:
12) (2,30°), (1,120°)
13) (-3,45°) and (7,105°).
14) (a, π/2) and (3a, π/6)
15) Prove that the points (0,0), (3,π/2) and (3,π/6) form an equilateral triangle.
Find the area of the triangles the coordinates of whose angular points are:
16) (1,30°), (2,60°) and (3,90°).
17) (-3, -30°),(5,150°) and (7,210°).
18) (-a, π/6),(a, π/2) and (2a, -2π/3).
Find the polar coordinates (drawing the figure in each case) of the points
19) find the current CM coordinates drawing a figure if each case at this point is point co-ordinate sir change to polar coordinates the equation transfer the cartesium coordinates the equation
₂₃₂₃₂₃₂₃₃₂₃₃₁₁₁₂₂₃₃
EQUATION OF LOCUS
By taking a number of solution sketch the Loki of the following equation a constant quantity Bing constant a constant quantity bring the point find the locus of a point is distance from the point 12 is equals to distance from the axis mind equation to the locals of the point which is always equal find the equation to the locus of a point reach more so that it distance from the exist of X is 3 times the distance from taxes are why resistance from the point is always distance from the excess of life the sum of equation of the distribution distance to 3 10 square of these distance from the point 02 is equals to prove is distance from the point 30 is three times if distance from 02 a diagram from the axis of X is always one heart is distance comparison a fixed point is a tape perpendicular distance from which state line and the point move so that is distance from the fixed point is always equal axis of coordinates drone through a fixed point and being parallel and perpendicular to the given line in this previous question with the first distance be always hard and always twice the second distance looking
THE STRAIGHT LINE
Find the equation to the straight line cutting of an intercept unity from the positive direction of the axis of a y an inclined at 45 degree to the axis of X cutting of an intersect by from the axis of a y and a beam equally inclined to the axis cutting of an interactive from the negative direction and inclined at angle to the taxi suffix find the equation of the straight line cutting of intercept 3 and 2 from the axis cutting of intersect minus 5 6 from the axis find the equation to the straight line is passes through the point 56 the line intercepts on the axis equivalent and both positive equal to the magnitude but opposite inside find the equation to the straight line which pass through the point 12 and cut off equal disstances from the 2x is find the equations to the straight line which passes through the given point and is such that given Find the equation to the straight line which passes through the point (-4,3) and you such that the person I could be doing Axis divided by the point in the ratio 53 raise the straight line is equations are find the equation to the straight line passing through the following pairs of points find the equation to the sides of the triangle the coordinates of the whose a angular points are respectively find the equation in the diagonals of the rectangle the equation of a sides are find the equation to the straight line which bisect the distance between the points and also by saves the distance between the points find the equation to the straight lines which go the origin and triceps
Find the angles between the pair of straight lines find the time zent of the angle between the lines will insert safe from the accessive respectively prove that the point 21 0 to 23 and 46 are the coordinates of the angular points of the parallelogram in find the angle between the equation to the straight line passing through the point 23 and perpendicular to the state line passing through the point passing through the point 43 and perpendicular to the straight line passing through 1327 find the equation to the straight line drawn at right angles to the straight line through the point where it is not the axis find the equation to the straight lineage biceps and his perpendicular to the straight line joining the points and prove that the equation to the straight line which passes through the point and is perpendicular to the straight line find the equation to the straight line passing through and respectivali perpendicular to the straight linesfind the equation to the straight lines is device internal and externally the lion joining 3754 ratio 4:7 which are perpendicular to the line through the point 34 address to straight lines each inclined of Y to the state lines find the equation and find also the area included by the three lines so that the equation to the straight line passing through the point 32 and in climate 60 to the line find the equation to the straight line with pass through the origin or indicated at 75 to this airline find the equation to the state language find the angle between two straight lines and also decoration in the two straight lines which passed through the point and make equal angle so which two given lines
Find the length of the perpendicular drawn from the point 45 upon the straight line the origin upon the straight line the point 34 upon the straight line the point up on the straight line find the length of the perpendicular from the origin upon straight line joining the two points is coordinates are so that the producer product of the perpendicular drawn from the two points up on the straight line if be the perpendicular from the origin up on the straight line is equation are proved that find the distance between two parallel straight lines what are the points on the access of xender so that the perpendicular fall from any point of the straight line up and two straight lines and equal to each other find the perpendicular distance from the origin of the perpendicular from the point 12 upong the straight line
Find the coordinates of the point of intersection of the straight lines pose equations are two straight lines cut the access distance and axis of why are distance respectively find the coordinates of their point of intersection find the distance of the point of intersection of the two statement from the straight line so that the perpendicular from the origin upon the straight line joining the points by the distance between them find the equation of two straight lines through the points on which perpendicular let fall from the point and each of the length joining the this perpendicular is find the point of intersection in the inclination in the two points find also the angle between them find the coordinates of the perpendicular LED fall the from the point 50 upon the sides of the triangle found by joining the three points 43 43 03 prove that the lines points are determined lying on the state line find the coordinates of the point of the intersection of the statement and determine also the angle of a reach the cut one another find angle between the two lines of the points 32prove that the points is coordinates are respectively prove that the following states of three lines meet in a point prove that the three state lines which equations are all emits in a point so also that third line bisect the angle between other two find the condition the straight lines may meet in a point find the coordinator of the auto centre of the triangle whose angular points are in any triangle prove that the bicycle angles meet in a point the medium line joining the each vertex of the middle point of the opposite sides in a point and the state line through the middle points of the sides perpendicular to the side find the equation to the straight line passing through the point 32 and the point of intersection of the line the point 29 and the intersection of the line the origin in the point of intersection probing that it biceps the angle between the origin in the point of intersection of the line the point and the intersection of the same two lines the intersection of the lines and parallel to the straight line the intersection of the lines and perpendicular section of the lines in the cutting of interceptthe intersection of the lines on the intersection of the lines if so the angular points of ultra and the straight lines be drawn parallel to the sides and if the intersect sim of this lines be join to opposite angular points of the triangle so that the joining lines are so obtained will meet in a plane find the equation to the statement passing through the point of intersection of lines passing through the origin parallel to the excess prove that the diagonals of the parallelogram found by four straight lines are at right angles to one another Prove the same property for the parallelogram besides are one side of a square is inclined to the axis of X at an angle and one operates extremities is at the origin prove that the equation to its diagonal where is the length of the side of the square find the equation to the straight line bisecting the angles between the followingfind the back sector of the angle between the straight lines find the equation to the bisector of the internal angles of the triangle equation is sides are respectively find the equation to the straight line passing through the foot of the perpendicular from the point of straight line and bisecting the angles between the perpendicular and the given square line find the direction in which state line must betron through the point 12 section with the line may be a distance from this point
Axis Bank in client at an angle 65 inclination to the axis of extract line whose equations are the axis being inclined at an angle of 120 find the tangent the angle between the two straight lines with oblique coordinates find the tangent of the angle between the straight line represent to straight line the right angle prove that the angle between the accessories prove that the straight line are at right angles whatever with the angle between find the equation to decide diagonals of the regular hexagon to a preside which meet in a bring the axis of the coordinates from each corner of the parallelogram perpendicular is drawn upon the diagonal is does not pass through that corner and these are produced to form another parallelogram so that is diagonal and regular to the sides of the first parallelogram and both have the same centre in the straight lines bag Axis Bank at an angle of 30 find the equation to the state line is passes through the point is perpendicular to the straight line find the length of the perpendicular one from the point 43 upon the straight line the angle between the axis been 60 find the equation 2 and the length of the perpendicular drawn from the point 11 upon the state line angle between axis in 120 the coordinator referred to access meeting at an angle are prove that the length of the straight line join the feet of the perpendicular form account
α β θ ₁ ₂ ₃ ₄ ₅ ₆ ₓ ᵢ ₙ ₋ ₊
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