Question

The locus of the points of trisection of the double ordinates of a parabola is a
(a) pair of lines
(b) circle
(c) parabola
(d) straight line

Answer 1

(c) parabola



Suppose PQ is a double ordinate of the parabola $y^2=4ax$.

Let R and S be the points of trisection of the double ordinates.

Let $\left(h,k\right)$ be the coordinates of R.

Then, we have:
OL = h and RL = k

$$\begin{gathered} \therefore RS=RL+LS=k+k=2k \\ \Rightarrow PR=RS=SQ=2k \\ \Rightarrow LP=LR+RP=k+2k=3k \end{gathered}$$

Thus, the coordinates of P are $\left(h,3k\right)$, which lie on $y^2=4ax$.

∴ $9k^2=4ah$

Hence, the locus of the point (h, k) is $9y^2=4ax$, i.e. $y^2=\left(\frac{4a}{9}\right)x$, which represents a parabola.