JEE Main 2025 Jan 28 Shift 2, Mathematics Q24: Introduction to Parabola

JEE Main2025Jan 28, Shift 2Mathematics

Let $A$ and $B$ be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0$ . If $d$ denotes the distance between $A$ and $B$ , and $a$ denotes the area of $△ S A B$ , where $S$ is the focus of the parabola $y^{2} = 4 x$ , then the value of $(a + d)$ is ____.

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Solution

The parabola $y^{2} = 4 x$ has vertex $(0, 0)$ and focus $(1, 0)$ . Reflecting in the line $x + y + 4 = 0$ :

vertex $(0, 0)$ maps to $(- 4, - 4)$ , focus $(1, 0)$ maps to $(- 4, - 5)$ , and the axis (the $x$ -axis) maps to the vertical line through these images, $x = - 4$ . So the reflected parabola has axis $x = - 4$ (vertical), vertex $(- 4, - 4)$ , focus $(- 4, - 5)$ , opening downward with semi-latus rectum 1 (latus rectum 4).

The line $y = - 5$ passes through the reflected focus and is therefore the latus rectum. Its chord length on the parabola is $∣4 p ∣ = 4$ , so $d = 4$ . The endpoints are $A = (- 6, - 5)$ and $B = (- 2, - 5)$ .

Triangle $S A B$ with $S = (1, 0)$ and base $A B$ on $y = - 5$ : base $= 4$ , height $= ∣0 - (- 5) ∣ = 5$ , area $a = \frac{1}{2} \cdot 4 \cdot 5 = 10$ .

$∴ a + d = 10 + 4 = 14$ .

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JEE Main 2025 Jan 28 Shift 2, Mathematics Q24: Introduction to Parabola

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