Question

Two circles with radii ' $r_{1}$ ' and ' $r_{2}$ ' $r_{1}>r_{2} \geq 2$, touch each other externally. If ' $\theta$ ' be the angle between the direct common tangents, then

• A. $\theta=\sin ^{-1}\left(\frac{r_{1}+r_{2}}{r_{1}-r_{2}}\right)$
• B. $\theta=2 \sin ^{-1}\left(\frac{r_{1}-r_{2}}{r_{1}+r_{2}}\right)$
• C. $\theta=\sin ^{-1}\left(\frac{r_{1}-r_{2}}{r_{1}+r_{2}}\right)$
• D. None of these

Answer 1

Answer: (B)

$\sin \alpha=\frac{r_{1}-r_{2}}{r_{1}+r_{2}}$

$\Rightarrow \theta=2 \sin ^{-1}\left(\frac{ r _{1}- r _{2}}{ r _{1}+ r _{2}}\right)$