Question

If tangent at $(1,2)$ to the circle $c_1: x^2+y^2=5$ intersects the circle $c_2: x^2+y^2=9$ at $A$ & $B$ and tangents at $A \& B$ to the second circle meet at point $C$, then the co-ordinates of $C$ are:

• A. $(4,5)$
• B. $\left(\frac{9}{15}, \frac{18}{5}\right)$
• C. $(4,-5)$
• D. $\left(\frac{9}{5}, \frac{18}{5}\right)$

Answer 1

Answer: (D)

Tangent at $(1,2)$ to the circle $x^2+y^2=5$
$x+2 y-5=0$
chord of contact from $C(h, k)$ to $x^2+y^2=9$
$h x+k y-9=0$

compare both equations $\frac{ h }{1}=\frac{ k }{2}=\frac{9}{5}$
$(h, k) \equiv\left(\frac{9}{5}, \frac{18}{5}\right)$