The mirror image of any point (α,β) with respect to line y=x is simply (β,α). Let (h,k) be the mirror image of a point on parabola y2=4ax Then, (k,h) will be the mirror image of (h,k) and it will lie on parabola. So, y2=4x ⇒x2=4y Hence, Locus is x2=4y⋅⋅⋅⋅⋅⋅⋅(i) For finding equation of tangent differentiate Eq. (i)w.r.t. x 2x=4
dy
dx
⇒
dy
dx
=
2x
4
=(
x
2
) ⇒(
dy
dx
)2,1=(
2
2
)=1 ⇒
y−1
x−2
=1⇒y−1=x−2 y=x−1 ∴ Equation of tangent ⇒y=x−1 ⇒x−y=1