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Q.
The equation of the parabola whose vertex is at $(2 ,-1 )$ and focus at $(2, - 3)$ is
Conic Sections
Solution:
Let $\left(\alpha, \beta\right)$ be any point on the directrix.
$A$ is the mid-point of $ZS$
Using mid point formula $\frac{\alpha+2}{2}=2$, $\frac{\beta-3}{2}=-1$
$\therefore \, \alpha=2$ and $\beta=-2+3=1$. So $Z\left(2,1\right)$
Since directrix is parallel to $x$-axis and passing through $\left(2,1\right)$, so its equation is $y - 1 = 0$
Since $PS = PM$ (for parabola)
$\therefore \, PS^{2}=PM^{2}$
$\Rightarrow \, \left(x-2\right)^{2}+\left(y+3\right)^{2}=\left(y-1\right)^{2}$
$\Rightarrow \, x^{2}-4x+8y+12=0$
