Question

The length of the tangents from any point on the circle $15 x^2+15 y^2-48 x+64 y=0$ to the two circles $5 x^2+$ $5 y^2-24 x+32 y+75=0$ and $5 x^2+5 y^2-48 x+64 y+300=0$ are in the ratio

• A. $ 1: 2$
• B. $2: 3$
• C. $3: 4$
• D. None of these

Answer 1

Answer: (A)

Let any point $P\left(x_1, y_1\right)$ to the circle $x^2+y^2-\frac{16 x}{5}+\frac{64 y}{15}=0$
$\Rightarrow x _1{ }^2+ y _1{ }^2-\frac{16 x _1}{5}+\frac{64 y _1}{15}=0$
Length of tangent from $P \left( x _1, y _1\right)$ to the circle are in ratio
$\frac{\sqrt{S_1}}{\sqrt{S_2}}=\frac{\sqrt{x_1^2+y_1^2-\frac{24}{5} x_1+\frac{32}{5} y_1+15}}{\sqrt{x_1^2+y_1^2-\frac{48}{5} x_1+\frac{64}{5} y_1+60}}$
$=\sqrt{\frac{\frac{16}{5} x_1-\frac{64}{15} y_1-\frac{24}{5} x_1+\frac{32}{5} y_1+15}{\frac{16}{5} x_1-\frac{64}{15} y_1-\frac{48}{5} x_1+\frac{64}{5} y_1+60}}$
$ =\sqrt{\frac{-24 x_1+32 y_1+225}{-96 x_1+128 y_1+900}}$
$=\sqrt{\frac{-24 x _1+32 y _1+225}{4\left(-24 x _1+32 y _1+225\right)}}=\frac{1}{2}$