Let PQ be a chord of the parabola and the midpoint of PQ be at (4, 1). Then, which of the following point lies on the line passing through the points P and Q? [2024]

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Solution

Q.
Let PQ be a chord of the parabola $y^{2} = 12 x$ and the midpoint of PQ be at (4, 1). Then, which of the following point lies on the line passing through the points P and Q [2024]

1 (2, –9)

2 $(\frac{3}{2}, - 16)$

3 $(\frac{1}{2}, - 20)$

4 (3, –3)

Ans.
(3)

$y^{2} = 12 x$

Chord PQ having mid-point $(x_{1}, y_{1}) \equiv (4, 1)$

Equation of chord PQ

$T = S_{1}$

$y y_{1} - 12 \frac{(x + x_{1})}{2} = y_{1}^{2} - 12 x_{1}$

$\Rightarrow y - 6 (x + 4) = 1 - 12 \times 4$

$\Rightarrow y - 6 x - 24 = - 47 \Rightarrow y - 6 x + 23 = 0$

From the options, point $(\frac{1}{2}, - 20)$ lies on the chord.

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