Question

If the lengths of the tangents drawn from the points $(1,2)$ to the circle $x^2+y^2+x+y-4=0$ and ${3x}^2+3y^2-x+y+k=0$ be in the ratio $4:3$, then $k=$

• A. $\frac{55}{4}$
• B. $\frac{7}{4}$
• C. $-\frac{55}{4}$
• D. $-\frac{7}{4}$

Answer 1

The correct option is C $-\frac{55}{4}$

Length of Tangent from point $(x_1,y_1)$ $=\sqrt{x_1^2+y_1^2+2g{x}_1+2f{y}_1+c}$

So, $\frac{4}{3}=\frac{\sqrt{1+4+1+2-4}}{\sqrt{3+12-1+2+k}}\Rightarrow \frac{16}{9}=\frac{4}{16+k}\Rightarrow k=\frac{-55}{4}$

Therefore, correct answer is $\frac{-55}{4}$.