Question

Tangents are drawn from points on the line $x+2y=8$ to the circle $x^2+y^2=16$. If locus of mid point of chord of contacts of the tangents is a circle $S$, then radius of circle $S$ will be

• A. $\sqrt{3}$
• B. $2$
• C. $\sqrt{5}$
• D. $2\sqrt{2}$

Answer 1

Answer: (C)

$\sqrt{5}$

Let $P(h,k)$ be mid points of chord, equation of chord of contact $hx+ky=h^2+k^2$ ...(1)
Let co-ordinate of $R(x_1,y_1)$, then equation of chord of contact $x{x}_1+y{y}_1=16$ ...(2)
Comparing (1) and (2)
$\frac{h}{x_1}=\frac{k}{y_1}=\frac{h^2+k^2}{16}$ $\Rightarrow x_1=\frac{16h}{h^2+k^2},y_1=\frac{16k}{h^2+k^2}$
As $R(x_1,y_1)$ will lie on line $x+2y=16$
Then $\frac{16h+32k}{h^2+k^2}=8$ $\Rightarrow h^2+k^2-2h-4k=0$
$\therefore$ Radius$=\sqrt{5}$