Two circles of radii `aa n db` touching each other externally, are inscribed in the area bounded by `y=sqrt(1-x^2)` and the x-axis. If `b=1/2,` then `a` is equal to (a)`1/4` (b) `1/8` (c) `1/2` (d) `1/(sqrt(2))`

Options

A. `1//4`
B. `1//8`
C. `1//2`
D. `1//sqrt(2)`

Answer

1

Explanation

Let the centers of the circle be `C_(1)` and `C_(2)`.
Then `C_(1)` is `(sqrt(1-2a),a)` and `C_(2)` is `(sqrt(1-2b),b)` . Now,
`C_(1)C_(2)=a+b=a+(1)/(2)`
or `1-2a+(a-(1)/(2))^(2)=(a+(1)/(2))^(2)`
or `a=(1)/(4)`