JEE Main 2025 Apr 7 Shift 1, Mathematics Q5: Introduction to Parabola

JEE Main2025Apr 7, Shift 1Mathematics

Let $P$ be the parabola, whose focus is $(- 2, 1)$ and directrix is $2 x + y + 2 = 0$ . Then the sum of the ordinates of the points on $P$ , whose abscissa is $- 2$ , is

A
$\frac{3}{2}$
B
$\frac{5}{2}$
C
$\frac{1}{4}$
D
$\frac{3}{4}$

Solution

By the focus-directrix definition, any point $(x, y)$ on the parabola satisfies

$(x + 2)^{2} + (y - 1)^{2} = (\frac{2 x + y + 2}{5})^{2}$

Multiply through by 5:

$5 [(x + 2)^{2} + (y - 1)^{2}] = (2 x + y + 2)^{2}$

Substitute $x = - 2$ to find the ordinates on the vertical line $x = - 2$ :

$5 (y - 1)^{2} = (- 4 + y + 2)^{2} = (y - 2)^{2}$

Expand: $5 (y^{2} - 2 y + 1) = y^{2} - 4 y + 4$

$\Rightarrow 5 y^{2} - 10 y + 5 = y^{2} - 4 y + 4$

$\Rightarrow 4 y^{2} - 6 y + 1 = 0$

Sum of roots $= y_{1} + y_{2} = \frac{6}{4} = \frac{3}{2}$ .

$∴$ Option (1).

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JEE Main 2025 Apr 7 Shift 1, Mathematics Q5: Introduction to Parabola

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