The figure shows a point P on the parabola y = x2 and the point Q where the perpendicular bisector of OP intersects the y‑axis. As the point P approaches the origin along the parabola, the limiting ordinate of Q is

Engineering

Mathematics

Chord of Parabola

Question

The figure shows a point P on the parabola y = x² and the point Q where the perpendicular bisector of OP intersects the y‑axis. As the point P approaches the origin along the parabola, the limiting ordinate of Q is

1/4

1/2

2/3

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line OP : y = tx with m = t

$M (\frac{t}{2}, \frac{t^{2}}{2})$

equation of QM, $y - \frac{t^{2}}{2} = - \frac{1}{t} (x - \frac{t}{2})$

put x = 0, $y - \frac{t^{2}}{2} = \frac{1}{2} \Rightarrow y = \frac{t^{2}}{2} + \frac{1}{2}$

Now $\underset{t \to 0}{Lim} y = \underset{t \to 0}{Lim} (\frac{t^{2}}{2} + \frac{1}{2}) = \frac{1}{2}$