If the X-coordinate of a point is twice the Y-coordinate and it is equidistant from R(2,-5) and S(-3,6) then the coordinate of point is___

(16,8)

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(8,16)

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$(\frac{1}{2}, 1)$

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Solution

The correct option is A
(16,8)

Given points are R(2,-5) and S(-3,6)

Let the required point be P(x,y) which is equidistant from both R & S.

$∴ \sqrt{(x - 2)^{2} + (y + 5)^{2}} = \sqrt{(x + 3)^{2} + (y - 6)^{2}}$

$∵$ Given that x=2y substitute in above equation

$\Rightarrow \sqrt{(2 y - 2)^{2} + (y + 5)^{2}} = \sqrt{(2 y + 3)^{2} + (y - 6)^{2}}$

$\Rightarrow (2 y - 2)^{2} + (y + 5)^{2} = (2 y + 3)^{2} + (y - 6)^{2}$

$\Rightarrow 4 y^{2} + 4 - 8 y + y^{2} + 25 + 10 y = 4 y^{2} + 9 + 12 y + y^{2} + 36 - 12 y$

$\Rightarrow 2 y + 20 = 45$

$\Rightarrow 2 y = 16$

$\Rightarrow y = 8$

$∴ x = 2 \times 8$

=16

$∴$ coordinates of P=(16,8)

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