JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Parabola

Reflecting a point $(x,y)$ in the line $x+y+4=0$ gives the image $(-y-4,-x-4)$. If $(X,Y)$ is the image of a point $(x,y)$ on $y^2=4x$, then $x=-Y-4$, $y=-X-4$, so

This is the mirror-image parabola.

Setting $Y=-5$:


So $A=(-2,-5)$, $B=(-6,-5)$, giving $d=|AB|=4$.

The focus of $y^2=4x$ is $S=(1,0)$. Since $AB$ is horizontal at height $y=-5$, the perpendicular distance from $S$ to line $AB$ is 5, so


$a+d=10+4=14$.