JEE Main 2025 (Online) 24th January Evening ShiftQuestion 1
If the equation of the parabola with vertex $V\left(\dfrac{3}{2},3\right)$ and the directrix $x+2y=0$ is $\alpha x^{2}+\beta y^{2}-\gamma xy-30x-60y+225=0$, then $\alpha+\beta+\gamma$ is equal to :
Parabola
Solution
The foot of $V$ on the directrix is $(0,0)$, so the focus is $(3,6)$. $5\left((x-3)^2+(y-6)^2\right)=(x+2y)^2$ gives $4x^2+y^2-4xy-30x-60y+225=0$: $\alpha+\beta+\gamma=4+1+4=9$.