Question

The locus of the points of trisection of the double ordinates of a parabola is a 

Options

• pair of lines 

•  circle

• parabola 

• straight line 

Answer 1

 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  

\[\therefore RS = RL + LS = k + k = 2k\]
\[ \Rightarrow PR = RS = SQ = 2k\]
\[ \Rightarrow LP = LR + RP = k + 2k = 3k\]

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

∴ \[9 k^2 = 4ah\] 

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