The equation of a circle is `x^2+y^2=4.` Find the center of the smallest circle touching the circle and the line `x+y=5sqrt(2)`

Options

A. `((7)/(2sqrt(2)),(7)/(2sqrt(2)))`
B. `((3)/(2),(3)/(2))`
C. `(-(7)/(2sqrt(2)),-(7)/(2sqrt(2)))`
D. none of these

Answer

A

Explanation

We have, OP=2 and OQ=5.
`:. CP=CQ=(3)/(2) rArr OC=2+(3)/(2)=(7)/(2)`
The equation of OC is `(x-0)/((1)/(sqrt(2)))=(y-0)/((1)/(sqrt(2)))=(7)/(2) rArr x=y=(7)/(2sqrt(2))`
Hence, the coordinates of C are `((7)/(2sqrt(2)),(7)/(2sqrt(2)))`