Question

The tangent to the circle $C_1:x^2+y^2-2x-1=0$ at the point $(2,1)$ cuts off a chord of length $4$ from a circle $C_2$ whose centre is $(3,-2)$. The radius of $C_2$ is

• A. $\sqrt{6}$
• B. $2$
• C. $\sqrt{2}$
• D. $3$

Answer 1

Answer: (A)

Equation of tangent on $C_1$ at $(2,1)$ is:

$2x+y-(x+2)-1=0$

$x+y=3$

If it cuts off the chord of the circle $C_2$ then the equation of the chord is:$x+y=3$

Distance of the chord from $(3,-2)$

$d=|\frac{3-2-3}{\sqrt{2}}|=\sqrt{2}$

Length of the chord is $l=4$

$r^2=\frac{l^2}{4+d^2}$ where $r$ is the radius of the circle.

$r^2=4+2=6\Rightarrow r=\sqrt{6}$