Advertisements
Advertisements
Question
The distances of point P (x, y) from the points A (1, - 3) and B (- 2, 2) are in the ratio 2: 3.
Show that: 5x2 + 5y2 - 34x + 70y + 58 = 0.
Advertisements
Solution
It is given that PA: PB = 2: 3
`"PA"/"PB" = (2)/(3)`
`"PA"^2/"PB"^2 = (4)/(9)`
`((x - 1)^2 + (y + 3)^2)/((x + 2)^2 + (y - 2)^2) = (4)/(9)`
`(x^2 + 1 -2x + y^2 + 9 + 6y)/(x^2 + 4 + 4x + y^2 + 4 - 4y) = (4)/(9)`
9(x2 - 2x + y2 + 10 + 6y) = 4(x2 + 4x + y2 + 8 - 4y)
9x2 - 18x + 9y2 + 90 + 54y = 4x2 + 16x + 4y2 + 32 - 16y
5x2 + 5y2 - 34x + 70y + 58 = 0
Hence, proved.
APPEARS IN
RELATED QUESTIONS
The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and R(–3, 6), find the coordinates of P.
If the point P(2, 2) is equidistant from the points A(−2, k) and B(−2k, −3), find k. Also find the length of AP.
Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), (8, – 2) and (2, – 2). Also, find its circum radius
The length of a line segment is of 10 units and the coordinates of one end-point are (2, -3). If the abscissa of the other end is 10, find the ordinate of the other end.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
Find the distances between the following point.
A(a, 0), B(0, a)
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
Find distance between point Q(3, –7) and point R(3, 3)
Solution: Suppose Q(x1, y1) and point R(x2, y2)
x1 = 3, y1 = –7 and x2 = 3, y2 = 3
Using distance formula,
d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `sqrt(square - 100)`
∴ d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `square`
Show that Alia's house, Shagun's house and library for an isosceles right triangle.
|
Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
- At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1, 2), B(4, 3) and C(6, 6)
- Check if the Goal keeper G(–3, 5), Sweeper H(3, 1) and Wing-back K(0, 3) fall on a same straight line.
[or]
Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ. - If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
