The line `x-y=1` intersects the parabola `y^2=4x` at `A` and `B` . Normals at `Aa n dB` intersect at `Cdot` If `D` is the point at which line `C D` is normal to the parabola, then the coordinates of `D` are `(4,-4)` (b) `(4,4)` `(-4,-4)` (d) none of these

Options

A. (4,-4)
B. (4,4)
C. (-4,-4)
D. none of these

Answer

B

Explanation

(2) solving the line y=x-1 and the parabola `y^(2)=4x`, we have
`(x-1)^(2)=x`
`orx^(2)-6x+1=0`
`orx=3pmsqrt(8)`
`:.y=2pmsqrt(8)`
Suppose point D is `(X_(3),y_(3))`. Then,
`y_(1)+y_(2)+y_(3)=0`
`or2+sqrt(8)+2-sqrt(8)+y_(3)=0`
`ory_(3)=-4`
Then `x_(3)=4`.
Therefore, the point is (4,-4).