JEE Main 2025 Jan 29 Shift 1, Mathematics Q3: Introduction to Parabola

JEE Main2025Jan 29, Shift 1Mathematics

Two parabolas have the same focus $(4, 3)$ and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points $A$ and $B$ , then $(A B)^{2}$ is equal to

A
$192$
B
$384$
C
$96$
D
$392$

Solution

Using the focus-directrix definition for a point $(x, y)$ on each parabola:

Parabola 1 (directrix $y = 0$ ): $(x - 4)^{2} + (y - 3)^{2} = y^{2} .$

Parabola 2 (directrix $x = 0$ ): $(x - 4)^{2} + (y - 3)^{2} = x^{2} .$

At an intersection point both hold, so $x^{2} = y^{2}$ . Combined with positivity of the LHS we get $x = y$ on the relevant branch.

Substituting $y = x$ into $(x - 4)^{2} + (x - 3)^{2} = x^{2}$ gives $x^{2} - 14 x + 25 = 0$ , whose roots $x_{1}, x_{2}$ satisfy $x_{1} + x_{2} = 14$ and $x_{1} x_{2} = 25$ .

Since $A = (x_{1}, x_{1})$ and $B = (x_{2}, x_{2})$ : $A B^{2} = 2 (x_{1} - x_{2})^{2} = 2 [(x_{1} + x_{2})^{2} - 4 x_{1} x_{2}] = 2 (196 - 100) = 192.$

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JEE Main 2025 Jan 29 Shift 1, Mathematics Q3: Introduction to Parabola

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