Parabola JEE Main PYQs, 2020 to 2025

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ 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 :

Let a line perpendicular to the line $2 x-y=10$ touch the parabola $y^2=4(x-9)$ at the point P. The distance of the point P from the centre of the circle $x^2+y^2-14 x-8 y+56=0$ is __________.

Let $P(\alpha, \beta)$ be a point on the parabola $y^2=4 x$. If $P$ also lies on the chord of the parabola $x^2=8 y$ whose mid point is $\left(1, \frac{5}{4}\right)$, then $(\alpha-28)(\beta-8)$ is equal to _________.

Let P₁ be a parabola with vertex (3, 2) and focus (4, 4) and P₂ be its mirror image with respect to the line x + 2y = 6. Then the directrix of P₂ is x + 2y = ____________.