Question

If the locus of the middle points of all chords of the parabola $y^2=12x$ passing through the vertex is the parabola $P$, then find the area bounded by the parabola $x^2=4y$ and $P$.

Answer 1

Answer: 8

$P\left(3t^2,6t\right)$
Mid-point of AP
$M\left(\frac{3}{2}{t}^2,3t\right)=M(h,k)$
$\therefore k^2=9t^2$
$h=\frac{3t^2}{2}=\frac{3k^2}{18}$
$\therefore \text{ locus is }P:y^2=6x$
Hence area bounded by the parabola $x^2=4y$ and $y^2=6x$ is $16\times (1)\times \frac{6}{4}\times \frac{1}{3}=\frac{96}{12}=8$