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प्रश्न
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.
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उत्तर
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.
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संबंधित प्रश्न
If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
Find the distance between the following pairs of points:
(2, 3), (4, 1)
If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
Find the distance of the following point from the origin :
(0 , 11)
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
Find distance between point A(–1, 1) and point B(5, –7):
Solution: Suppose A(x1, y1) and B(x2, y2)
x1 = –1, y1 = 1 and x2 = 5, y2 = – 7
Using distance formula,
d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ d(A, B) = `sqrt(square +[(-7) + square]^2`
∴ d(A, B) = `sqrt(square)`
∴ d(A, B) = `square`
Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.
Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
The distance between the points A(0, 6) and B(0, –2) is ______.
What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?
