Question

The lengths of tangents from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-\lambda =0$ are in the ratio 5:4 then the value of $\lambda$ equals

• A. $\frac{4}{17}$
• B. $\frac{21}{4}$
• C. $\frac{108}{25}$
• D. $\frac{12}{25}$

Answer 1

The correct option is C $\frac{108}{25}$

Let $S:x^2+y^2+x+y-4=0$ and $S^{\prime }:x^2+y^2-\frac{x}{3}-\frac{y}{3}-\frac{\lambda }{3}=0$
Length of tangents from $(1,2)$ to $S,S^{\prime }$ are in the ratio $5:4$
$\Rightarrow \frac{\sqrt{S_{11}}}{\sqrt{S_{11}^{\prime }}}=\frac{5}{4}$
$\Rightarrow \frac{S_{11}}{S_{11}^{\prime }}=\frac{25}{16}$
$\Rightarrow \frac{4\times 3}{(12-\lambda )}=\frac{25}{16}$
$\therefore \lambda =\frac{108}{25}$
Hence, option C.