If the equation of the parabola with vertex V (3/2, 3 ) and the directrix x+2 y=0 is α x 2 +β y 2 -γ x y-30 x-60 y+225=0 , then α+β+γ is equal to:

Mathematics · Parabola

JEE Main 2025 — 24 January, Evening Shift — Question 18

If the equation of the parabola with vertex $V\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is

$\alpha x^{2}+\beta y^{2}-\gamma x y-30 x-60 y+225=0$ , then $\alpha+\beta+\gamma$ is equal to:

Equation of axis $y-3=2\left(x-\frac{3}{2}\right)$

y-2 x=0

foot of directrix

y-2 x=0

& $2 y+x=0$

\Rightarrow(0,0)

\text{Focus } = (3, 6)

PS^2 = PM^2

(x-3)^2 + (y-6)^2 = \left( \frac{x+2y}{\sqrt{5}} \right)^2

4x^2 + y^2 - 4xy - 30x - 60y + 225 = 0

\Rightarrow \alpha = 4, \beta = 1, \gamma = 4 \Rightarrow \alpha + \beta + \gamma = 9

Answer key and solution verified before publishing.

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