If `C_(1): x^(2)+y^(2) =(3+2sqrt(2))^(2)` be a circle. PA and PB are pair of tangents on `C_(1)` where P is any point on the director circle of `C_(1)`, then the radius of the smallest circle which touches `C_(1)` externally and also the two tangents PA and PB is

Options

A. 1
B. 2
C. 3
D. 4

Answer

A

Explanation

Let Q be the centre of the given circle and R be the centre of the smallest circle touching the given circle externally. Then

`AQ = 3 +2 sqrt(2)`
`PQ = 3sqrt(2) +4`
Let radius of smaller circle is r.
`rArr 3 sqrt(2) +4 = QR +RP =3 +2 sqrt(2) +r +r sqrt(2) ( :' /_APQ = (pi)/(4))`
`rArr (sqrt(2)+1) = r (sqrt(2)+1) = r= 1`