·If the normals of the parabola `y^2=4x` drawn at the end points of its latus rectum are tangents to the circle `(x-3)^2 (y+2)^2=r^2` , then the value of `r^2` is

Answer

2

Explanation

End points of latusrectum are `(a +- 2a)` i.e. `(1 +- 2)`
Equation of normal at `(x_(1), y_(1))` is
`(y - y_(1))/(x - x_(1)) = (y_(1))/(2a)`
i.e., `(y - 2)/(x - 1) = - (2)/(2)` and `(y + 2)/(x - 1) = (2)/(3)`
`implies x + y + 3`
and `x - y = 3`
Which is tangent to `(X - 3)^(2) + (y + 2)^(2) = r^(2)`

`:.` Length of perpendicular from centre = Radius
`implies (|3 - 2 - 3|)/(sqrt(1^(2) + 1^(2))) = r`
`:. r^(2) = 2`