question_answer

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)             \[\frac{{{r}_{1}}{{r}_{2}}}{\sqrt{(r_{1}^{2}+r_{2}^{2})}}\]

Correct Answer: D

Solution :

                    In \[\Delta OPC\], \[\cos \theta =\frac{r}{{{r}_{1}}}\]                                 In \[\Delta O'PC\], \[\sin \theta =\frac{r}{{{r}_{2}}}\]                                 As, \[{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1\]                                                                          \ \[{{\left( \frac{r}{{{r}_{1}}} \right)}^{2}}+{{\left( \frac{r}{{{r}_{2}}} \right)}^{2}}=1\]Þ \[r=\frac{{{r}_{1}}{{r}_{2}}}{\sqrt{r_{1}^{2}+r_{2}^{2}}}\].