If y=mx+c is the normal at a point on the parabola y^2=8x whose focal distance is 8 units,…

Question

If $y=mx+c$ is the normal at a point on the parabola $y^2=8x$ whose focal distance is $8$ units, then $|c|$ is equal to:

Accepted Answer

$a=2$. Focal distance $=x_1+a=8\Rightarrow x_1=6=2t^2\Rightarrow t^2=3,\ t=\sqrt3$.

Normal at parameter $t$: $y=-tx+2at+at^3=-tx+at(2+t^2)$, so
$$|c|=at(2+t^2)=2\sqrt3(2+3)=10\sqrt3.$$

Solutions · 1