Question

A circle $C$ of unit radius lies in the first quadrant and touches both the axes. The circle $C_1$ touches both the axes and intersects $C$ such that the common chord is the longest. Its radius is

• A. 2
• B. 3
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

Answer: (B), (D)

The correct options are
B 3
D $\frac{1}{3}$

Radius of circle C is 1 and centre is $(1,1)$.
So, equation of circle C is:
$x^2+y^2-2(x+y)+1=0$

Let the radius of circle $C_1$ be $r$, so, its centre is $(r,r)$.
Hence, equation of circle $C_2$ is:
$x^2+y^2-2r(x+y)+r^2=0$

Equation of the common chord is $C-C_1=0$
$\Rightarrow 2(r-1)(x+y)=r^2-1$
$\Rightarrow 2(x+y)=1+r$ ... (i)

For the chord to be longest, either $(1,1)$ or $(r,r)$ lies on (i).
If $(1,1)$ lies on (i), then $2(1+1)=1+r\Rightarrow r=3$
If $(r,r)$ lies on (i), then $2(r+r)=1+r\Rightarrow r=\frac{1}{3}$