Question

The locus of a point which moves so that the ratio of the length of the tangents to the circles $x^2+y^2+4x+3=0$ and $x^2+y^2-6x+5=0$ is $2:3$, is

• A. $5x^2+5y^2-60x+7=0$
• B. $5x^2+5y^2+60x-7=0$
• C. $5x^2+5y^2-60x-7=0$
• D. $5x^2+5y^2+60x+7=0$

Answer 1

Answer: (D)

$5x^2+5y^2+60x+7=0$

Length of tangent at the point $P(x_1,y_1)$ 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 at the point $P(x_1,y_1)$ to the circle
$x^2+y^2-6x+5=0$
is $\sqrt{x_1^2+y_1^2-6x_1+5}$
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
$5x^2+5y^2+60x+7=0$