If the chord joining the points and on the parabola subtends a right angle at the vertex of the parabola, then is equal to [2026]

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Solution

Q.
If the chord joining the points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the parabola $y^{2} = 12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2} - y_{1} y_{2}$ is equal to [2026]

1 284

2 292

3 280

4 288

Ans.
(4)

$(x_{1} y_{1}) = (3 t_{1}^{2}, 6 t_{1}) and (x_{2} y_{2}) = (3 t_{2}^{2}, 6 t_{2})$

$t_{1} t_{2} = - 4$

$x_{1} x_{2} = 9 {(t_{1} t_{2})}^{2}, y_{1} y_{2} = 36 t_{1} t_{2}$

$x_{1} x_{2} - y_{1} y_{2} = 9 (16) - 36 (- 4)$

$= 144 + 144$

$= 288$

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