JEE Main 2026 — Jan 22, Shift 1 — Mathematics solution

JEE Main2026Jan 22, Shift 1Mathematics

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

A
288
B
280
C
284
D
292

Solution

$y^{2} = 12 x$ has $4 a = 12$ , so $a = 3$ . Parametric form: $P_{i} = (3 t_{i}^{2}, 6 t_{i})$ .

For a chord subtending a right angle at the vertex $(0, 0)$ : $O P_{1} \cdot O P_{2} = 0$ , i.e. $9 t_{1}^{2} t_{2}^{2} + 36 t_{1} t_{2} = 0$ , so $t_{1} t_{2} (t_{1} t_{2} + 4) = 0$ . Hence $t_{1} t_{2} = - 4$ .

$x_{1} x_{2} = 9 (t_{1} t_{2})^{2} = 9 (16) = 144$ .

$y_{1} y_{2} = 36 (t_{1} t_{2}) = 36 (- 4) = - 144$ .

$x_{1} x_{2} - y_{1} y_{2} = 144 - (- 144) = 288$ .

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