Two circles with radii `a and b` touch each other externally such that `theta` is the angle between the direct common tangents, `(a > bgeq2)` . Then prove that `theta=2sin^(-1)((a-b)/(a+b))` .

Options

A. `theta = sin ^(-1) ((r_(1)+r_(2))/(r_(1)-r_(2)))`
B. `theta = 2 sin^(-1) ((r_(1)-r_(2))/(r_(1)+r_(2)))`
C. `theta=sin^(-1)((r_(1)-r_(2))/(r_(1)+r_(2)))`
D. none of these

Answer

B

Explanation

In `DeltaC_(1) LC_(2)`, we have
`sin. (theta)/(2)=(C_(1)L)/(C_(1)C_(2))=(r_(1)-r_(2))/(r_(1)+r_(2)) rArr theta = 2 sin ^(-1) ((r_(1)-r_(2))/r_(1)+r_(2))`