Question

If two spheres of radii $r_{1}$ and $r_{2}$ cut orthogonally, then the radius of the common circle is

• A. $r_{1}r_{2}$
• B. $\sqrt{(r_{1}^{2} + r_{2}^{2}})$
• C. $r_{1}r_{2}\sqrt{(r_{1}^{2} + r_{2}^{2})}$
• D. $\frac{r_{1}r_{2}}{\sqrt{(r_{1}^{2} + r_{2}^{2})}}$

Answer 1

Answer: (D)

$\frac{r_{1}r_{2}}{\sqrt{(r_{1}^{2} + r_{2}^{2})}}$

Answer 2

In $\triangle O P C$, $\cos \theta = \frac { r } { r _ { 1 } }$

In , $\sin \theta = \frac { r } { r _ { 2 } }$

As, $\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1$

∴ $\left( \frac { r } { r _ { 1 } } \right) ^ { 2 } + \left( \frac { r } { r _ { 2 } } \right) ^ { 2 } = 1$⇒ $r = \frac { r _ { 1 } r _ { 2 } } { \sqrt { r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } } }$ .