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$A (- 9, 0), B (- 1, 0)$ are two points. If $P$ is a point such that $P A : P B = 3 : 1$ , then the locus of $P$ is

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$x^{2} + y^{2} = 9$

$x^{2} + y^{2} + 9 = 0$

$x^{2} - y^{2} = 9$

$x^{2} - y^{2} + 9 = 0$

answer is A.

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Detailed Solution

Given points $A (- 9, 0), B (- 1, 0) & P A : P B = 3 : 1$

Let the locus of the point $P (x_{1}, y_{1})$ then

$\frac{P A}{P B} = \frac{3}{1}$

$P A^{2} = 9 P B^{2}$

${(x_{1} + 9)}^{2} + {(y_{1} - 0)}^{2} = 9 [{(x_{1} + 1)}^{2} + y^{2}]$

$x_{1}^{2} + 18 x_{1} + 81 + y_{1}^{2} = 9 x_{1}^{2} + 18 x_{1} + 9 + 9 y^{2}$

$8 x_{1}^{2} + 8 y_{1}^{2} - 72 = 0$

$x_{1}^{2} + y_{1}^{2} - 9 = 0$

The locus of $P$ is $x^{2} + y^{2} = 9$

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