$A = (1, - 1)$ locus of $B$ is $x^{2} + y^{2} = 16$ . lf $P$ divides $A B$ in the ratio $3 : 2$ then locus of $P$ is

(x - 2)^{2} + (y - 3)^{2} = 4

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(x + 1)^{2} + (y - 2)^{2} = 4

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(x - 3)^{2} + (y - 2)^{2} = 4

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(5 x - 2)^{2} + (5 y + 2)^{2} = 144

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Solution

The correct option is C $(5 x - 2)^{2} + (5 y + 2)^{2} = 144$
let P (h,k) then PA/PB = 3/2
using ration relation we get the co-ordinates of the point P , x1 and y1 be the point B on circle
therefore its also satisfy the circle equation
$h = \frac{3 x_{1} + 2}{5}, k = \frac{3 y_{1} - 2}{5}$
$x_{1}^{2} + y_{1}^{2} = 16$
$x_{1} = \frac{5 h - 2}{3}, y_{1} = \frac{5 k + 2}{3}$
substitute the values of x1 and y1 we get,
$\frac{(5 h - 2)^{2}}{9} + \frac{(5 k + 2)^{2}}{9} = 16$
$(5 x - 2)^{2} + (5 y + 2)^{2} = 144$

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