Let PQ be a double ordinate of the parabola y^2=-4x, where P lies in the second quadrant.

Question

Let $PQ$ be a double ordinate of the parabola $y^2=-4x$, where $P$ lies in the second quadrant. If $R$ divides $PQ$ in the ratio $2:1$, then the locus of $R$ is:

Accepted Answer

Points on $y^2=-4x$: for $x=-t^2$, $y=\pm2t$. With $P$ in the 2nd quadrant, $P=(-t^2,2t)$, $Q=(-t^2,-2t)$.

$R$ divides $PQ$ in ratio $2:1$ (from $P$ to $Q$):
$$R=\left(\frac{2(-t^2)+1(-t^2)}{3},\ \frac{2(-2t)+1(2t)}{3}\right)=\left(-t^2,\ -\frac{2t}{3}\right).$$
So $t=-\dfrac{3y}{2}$, $t^2=\dfrac{9y^2}{4}=-x\Rightarrow 9y^2=-4x.$

Solutions · 1