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प्रश्न
If P (x , y ) is equidistant from the points A (7,1) and B (3,5) find the relation between x and y
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उत्तर
Let the point P(x, y) be equidistant from the points A(7, 1) and B(3, 5) Then
PA = PB
`⇒ PA^2 = PB^2`
`⇒(x-7)^2 +(y-1)^2 = (x-3)^2 +(y-5)^2`
`⇒ x^2 +y^2 -14x-2y +50 = x^2 +y^2 -6x -10y +34`
`⇒ 8x -8y = 16 `
`⇒ x-y =2`
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संबंधित प्रश्न
If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.
Find the value of x, if the distance between the points (x, – 1) and (3, 2) is 5.
Determine if the points (1, 5), (2, 3) and (−2, −11) are collinear.
If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
Find the distance between the following point :
(sec θ , tan θ) and (- tan θ , sec θ)
Find the point on the x-axis equidistant from the points (5,4) and (-2,3).
Find the coordinate of O , the centre of a circle passing through P (3 , 0), Q (2 , `sqrt 5`) and R (`-2 sqrt 2` , -1). Also find its radius.
Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram.
From the given number line, find d(A, B):
A point P lies on the x-axis and another point Q lies on the y-axis.
Write the ordinate of point P.
Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.
Points A (-3, -2), B (-6, a), C (-3, -4) and D (0, -1) are the vertices of quadrilateral ABCD; find a if 'a' is negative and AB = CD.
The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, -1) and the length of its diameter is 20 unit.
Use distance formula to show that the points A(-1, 2), B(2, 5) and C(-5, -2) are collinear.
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
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:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
∆ABC with vertices A(–2, 0), B(2, 0) and C(0, 2) is similar to ∆DEF with vertices D(–4, 0), E(4, 0) and F(0, 4).
What is the distance of the point (– 5, 4) from the origin?
Find the distance between the points O(0, 0) and P(3, 4).
