Question

If the lengths of the 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 $4:3,$ then the value of $\lambda$ is

• A. $\frac{21}{2}$
• B. $\frac{21}{4}$
• C. $\frac{21}{5}$
• D. $\frac{21}{11}$

Answer 1

Answer: (B)

Length of tangent from the point (1, 2) to the circle $x^2+y^2+x+y-4=0$ is $\sqrt{1+4+1+2-4}=2$ Similarly, length of tangent from the point (1,2) to the circle $3x^2+3y^2-x-y-\lambda =0$ or $x^2+y^2-\frac{x}{3}-\frac{y}{3}-\frac{\lambda }{3}=0$ is $\sqrt{1+4-\frac{1}{3}-\frac{2}{3}-\frac{\lambda }{3}}=\sqrt{4-\frac{\lambda }{3}}$ But given that ratio of lengths of tangents to two circles is $4:3$ . $\therefore$ $\frac{2}{\sqrt{4-\frac{\lambda }{3}}}=\frac{4}{3}$ $\Rightarrow$ $2\sqrt{4-\frac{\lambda }{3}}=3$ On squaring, $4\left(4-\frac{\lambda }{3}\right)=9$ $\Rightarrow$ $4-\frac{\lambda }{3}=\frac{9}{4}$ $\Rightarrow$ $\frac{\lambda }{3}=4-\frac{9}{7}=\frac{16-9}{4}=\frac{7}{4}$ $\Rightarrow$ $\lambda =\frac{21}{4}$