A circle touches the parabola `y^2= 4x` at `(1, 2)` and also touches its directrix. The y-coordinates of the point of contact of the circle and the directrix is-

Options

A. `sqrt(2)`
B. `2`
C. `2sqrt(2)`
D. `4`

Answer

C

Explanation

`y^(2) = 4x`
`2y(dy)/(dx) = 4`
`m_(T) = (2)/(y) = (2)/(2) = 1`
Circle `rarr S + lambdaL = 0`
`(x+1)^(2) + (y-alpha)^(2) + lambda(x+1) = 0"….."(1)`
differentiate
` 2(x+1)^(2) + 2(y-alpha) (dy)/(dx) + lambda = 0`
`x = 1, y = 2`
`4 + 2(2-alpha) m_(T) + lambda = 0`
`lambda = 2alpha - 8 "......"(2)`
`(1,2)` satisfies eq. (1)
`2^(2) + (2-alpha)^(2) + 2lambda = 0`
`alpha^(2) - 4alpha + 8 + 2(2alpha - 8) = 0`
`alpha^(2) = 8`
`alpha = 2sqrt(2)`