question_answer

Two circles of radii 4 cm and 1 cm touch each other externally and \[\theta \] is the angle contained by their direct common tangents. Then sin \[\theta \] is equal to -

A) \[\frac{24}{25}\]                                   

B) \[\frac{12}{25}\]

C) \[\frac{3}{4}\]                          

D) None of these

Correct Answer: A

Solution :

In \[\Delta {{C}_{1}}{{C}_{2}}Q\]

\[\sin \left( \frac{\theta }{2} \right)=\frac{{{C}_{1}}Q}{{{C}_{1}}{{C}_{2}}}=\frac{3}{5}\]

\[\cos \left( \frac{\theta }{2} \right)=\frac{{{C}_{1}}Q}{{{C}_{1}}{{C}_{2}}}=\frac{4}{5}\]

\[\therefore \,\,\sin \theta =2\sin \left( \frac{\theta }{2} \right)\cos \left( \frac{\theta }{2} \right)\]

\[=2\,\,\left( \frac{3}{5} \right)\,\,\left( \frac{4}{5} \right)=\frac{24}{25}\]