Question

Let two circles having radii $r_{1}$ and $r_{2}$ are orthogonal to each other. If the length of their common chord is $k$ times the square root of harmonic mean between the squares of their radii, then $k^{4}$ is equal to

• A. $13$
• B. $7$
• C. $4$
• D. $2$

Answer 1

Answer: (C)

Area of $\Delta ABC$
$=\frac{1}{2}r_{1}r_{2}=\frac{1}{2}\left(\frac{l}{2}\right)\sqrt{r_{1}^{2} + r_{2}^{2}}$
$\Rightarrow l=\frac{2 r_{1} r_{2}}{\sqrt{r_{1}^{2} + r_{2}^{2}}}$
$\Rightarrow l=\sqrt{2}\sqrt{\frac{2 r_{1}^{2} r_{2}^{2}}{r_{1}^{2} + r_{2}^{2}}}\Rightarrow k=\sqrt{2}$
$\Rightarrow k^{4}=4$