Question

Two perpendicular chords are drawn from the origin 'O' to the parabola $y=x^2$, which meet the parabola at P and Q Rectangle POQR is completed. Find the locus of vertex R.

Answer 1

Parabola: $y=x^2..........\left(1\right)$

Let $P$ be $(l,l^2)$ and $Q(m,m^2)$

Let $R$ be $(h,k)$

$=>m_2=$Slope of$OQ=\frac{m^2-0}{m-0}=m$

$=>m_1=$slope of$OP=\frac{l^2-0}{l-0}=l$

as $OQ$ perpendicular $OP$ $=>m_1{m}_2=-1$

$=>ml=-1....................\left(2\right)$

Point of intersection of diagonals $N$ is midpoint of $RO$ and $PQ$ which coincides at $N.$

So, Midpoint of $PQ=N:(\frac{l=m}{2},\frac{m^2+l^2}{2})$

Midpoint of $RO=N:(\frac{h+0}{2},\frac{k+0}{2})=>N:(\frac{h}{2},\frac{k}{2})$

Comparing them, $h=l+m,k=m^2+l^2$

$=>k=(m+l{)}^2-2ml$

$=>k=h^2-2(-1)$

$=>h^2=k-2$

For locus, $h⟶x,k⟶y$

$=>x^2=y-2.$