Explanation

(c)
Let $P(x_{1},y_{2})$ be any point outside the circle. Length of tangent to the circle $x^{2} + y^{2} + 4x + 3 = 0$ is
$\sqrt{x_{1}^{2} + y_{1}^{2} + 4x_{1} + 3}$
And length of tangent of the circle $x^{2} + y^{2} - 6x + 5 = 0$ is
$\sqrt{x_{1}^{2} + y_{1}^{2} - 6x_{1} + 5}$
$\therefore$ According to question,
$\frac{\sqrt{x_{1}^{2} + y_{1}^{2} + 4x_{1} + 3}}{\sqrt{x_{1}^{2} + y_{1}^{2} - 6x_{1} + 5}} = \frac{2}{3}$
$\Rightarrow 9x_{1}^{2} + 9y_{1}^{2} + 36x_{1} + 27 - 4x_{1}^{2} - 4y_{1}^{2} + 24x_{1} - 20 = 0$
$\Rightarrow \ \ 5x_{1}^{2} + 5y_{1}^{2} + 60x_{1} + 7 = 0$
$\therefore$ Locus of point is $5x^{2} + 5y^{2} + 60x + 7 = 0$