The radius of the circle touching the parabola `y^2=x` at (1, 1) and having the directrix of `y^2=x` as its normal is `(5sqrt(5))/8` (b) `(10sqrt(5))/3` `(5sqrt(5))/4` (d) none of these

Options

A. `5sqrt(5)//8`
B. `10sqrt(5)//3`
C. `5sqrt(5)//4`
D. none of these

Answer

C

Explanation

(3) The equation of normal at P (1,1) is
`y-1=-(x-1)`

`ory+2x=3` (1)
The directrix of parabola `y^(2)=x` is
`x=-(1)/(4)` (2)
The center of the circle is the intersection of two normals to the circle, i.e., (1) and (2). Therefore, the center is (-1/4,7/2).
Hence, the radius of the circle is
`sqrt((1+(1)/(4))^(2)+(1-(7)/(2))^(2))=sqrt((25)/(16)+(25)/(4))=(5sqrt(5))/(4)`