Question

The equation of the parabola whose vertex is at $(2,-1)$ and focus at $(2,-3)$ is

• A. $x^2+4x-8y-12=0$
• B. $x^2-4x+8y+12=0$
• C. $x^2+8y=12$
• D. $x^2-4x+12=0$

Answer 1

Answer: (B)

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)
$\therefore P{S}^2=P{M}^2$
$\Rightarrow {\left(x-2\right)}^2+{\left(y+3\right)}^2={\left(y-1\right)}^2$
$\Rightarrow x^2-4x+8y+12=0$