The equation to the locus of a point P for which the distance from P to (0, 5) is double the distance from P to y-axis is

3 x^{2} + y^{2} + 10 y - 25 = 0

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3 x^{2} - y^{2} + 10 y + 25 = 0

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3 x^{2} - y^{2} + 10 y - 25 = 0

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3 x^{2} + y^{2} - 10 y - 25 = 0

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Solution

The correct option is C $3 x^{2} - y^{2} + 10 y - 25 = 0$
Let $P (x, y)$ be an arbitrary point in $X - Y$ plane.
Therefore distance of P from $A (0, 5)$ is
$P A = \sqrt{x^{2} + (y - 5)^{2}}$ ...(i)
And distance of P from y axis.
$d_{y} = | x |$
It is given that
$2 d_{y} = P A$
Or
$2 | x | = \sqrt{x^{2} + (y - 5)^{2}}$
$4 x^{2} = x^{2} + (y - 5)^{2}$
$3 x^{2} - (y - 5)^{2} = 0$
$3 x^{2} - y^{2} + 10 y - 25 = 0$

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