Find the equation of the parabola if

the focus is at (0, −3) and the vertex is at (0, 0)

#### Solution

In a parabola, the vertex is the mid-point of the focus and the point of intersection of the axis and the directrix.

Let (*x _{1}, y_{1}*) be the coordinates of the point of intersection of thIt is given that the vertex and the focus of a parabola are (0, 0) and (0, −3), respectively.

Thus, the slope of the axis of the parabola cannot be defined.

Slope of the directrix = 0

Let the directrix intersect the axis at

*K*(

*r*,

*s*). e axis and directrix.

∴ \[\frac{r + 0}{2} = 0, \frac{s - 3}{2} = 0\]

\[ \Rightarrow r = 0, s = 3\]

∴ Required equation of directrix:

\[y = 3\]

Let *P *(*x*, *y*) be any point on the parabola whose focus is *S* (0, −3) and the directrix is \[y = 3\]

Draw *PM *perpendicular to \[y = 3\]

Then, we have:

\[SP = PM\]

\[ \Rightarrow S P^2 = P M^2 \]

\[ \Rightarrow \left( x - 0 \right)^2 + \left( y + 3 \right)^2 = \left( \frac{y - 3}{\sqrt{1}} \right)^2 \]

\[ \Rightarrow x^2 + y^2 + 6y + 9 = y^2 - 6y + 9\]

\[ \Rightarrow x^2 = - 12y\]