Question

The mirror image of the parabola $y^2=4x$ with respect to the tangent to the parabola at the point $(1,2)$ is

• A. ${(x-1)}^2=4(y-2)$
• B. ${(x-3)}^2=4(y+2)$
• C. ${(x+1)}^2=4(y-1)$
• D. ${(x-1)}^2=4(y-1)$

Answer 1

The correct option is C ${(x+1)}^2=4(y-1)$

$y^2=4x$

Point $(t^2,2t)$

Tangent at point is $y-2=1(x-1)$

$x-y+1=0$

Mirror image

(h, k) of point $(t^2,2t)$ in $x-y+1=0$

$\frac{h-t^2}{1}=\frac{k-2t}{-1}=\frac{-t^2+2t-1}{1}$

$h=t^2-t^2+2t-1$ $k=2t+t^2-2t+1$

$h=2t-1$ $k=t^2+1$

$t=\frac{h+1}{2}$ $k-1={\left(\frac{h+1}{2}\right)}^2$

i.e., $4(k-1)=(h+1{)}^2$

i.e., $(x+1{)}^2=4(y-1)$.