The locus of a point which moves such that its distance from the point $(0, 0)$ is twice its distance from the $Y$ -axis, is

x^{2} - y^{2} = 0

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x^{2} - 3 y^{2} = 0

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3 x^{2} - y^{2} = 0

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None of these

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Solution

The correct option is D $3 x^{2} - y^{2} = 0$
By hypothesis,
$| O P | = 2 | P M |$
$\Rightarrow \sqrt{x_{1}^{2} + y_{1}^{2}} = 2 | x_{1} |$
On squaring both sides, we get
$x_{1}^{2} + y_{1}^{2} = 4 x_{1}^{2} \Rightarrow 3 x_{1}^{2} - y_{1}^{2} = 0$
$∴$ Locus of the point is $3 x^{2} - y^{2} = 0$ .

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