MCQ

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

#### Options

pair of lines

circle

parabola

straight line

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#### Solution

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.

Is there an error in this question or solution?

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