Question

If the focus of a parabola is $(0,-3)$ and its directrix is $y=3$, then its equation is

• A. $x^2=-12y$
• B. $x^2=12y$
• C. $y^2=-12x$
• D. $y^2=12x$

Answer 1

The correct option is A $x^2=-12y$

Let $(h,k)$ be any point on the curve.

Distance of this point from the focus$=\sqrt{(h-0{)}^2+(k-(-3){)}^2}$

Distance of point $(h,k)$ from the directrix $=\frac{k-3}{1}=k-3$

Any point on the parabola is equidistant from its focus and dirextrix.

$⟹\sqrt{(h-0{)}^2+(k-(-3){)}^2}=k-3$

$⟹\sqrt{h^2+(k+3{)}^2}=k-3$

Squaring the above equation, we get

$⟹h^2+(k+3{)}^2=(k-3{)}^2$

$⟹h^2+k^2+6k+9=k^2-6k+9$

$⟹h^2=-12k$

Subtitute $k=y$ and $h=x$, we get

$⟹x^2=-12y$

So, the answer is option (A)