Question

If circles with radii $a$ units and $b$ units touch each other externally and the angle between their direct common tangents is $\theta$, where $a>b\ge 2$, then the value of $\sin \theta$ is

• A. $\frac{2(a-b)\left(\sqrt{ab}\right)}{(a+b{)}^2}$
• B. $\frac{2(a+b)\left(\sqrt{ab}\right)}{(a-b{)}^2}$
• C. $\frac{4(a-b)\left(\sqrt{ab}\right)}{(a+b{)}^2}$
• D. $\frac{4(a+b)\left(\sqrt{ab}\right)}{(a-b{)}^2}$

Answer 1

The correct option is C $\frac{4(a-b)\left(\sqrt{ab}\right)}{(a+b{)}^2}$


From the given conditions, we get
$\sin \alpha =\frac{a-b}{a+b}\cos \alpha =\frac{2\sqrt{ab}}{a+b}$

Now,
$\theta =2\alpha \Rightarrow \sin \theta =\sin 2\alpha \Rightarrow \sin \theta =2\sin \alpha \cos \alpha \Rightarrow \sin \theta =\frac{a-b}{a+b}\times \frac{4\sqrt{ab}}{a+b}\therefore \sin \theta =\frac{4(a-b)\left(\sqrt{ab}\right)}{(a+b{)}^2}$