Question

Find the equation of the parabola if 

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

Answer 1

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

Let (x₁, y₁) 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: 

\[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 0 \right)^2 + \left( y + 3 \right)^2 = \left( \frac{y - 3}{\sqrt{1}} \right)^2 \]
\[ \Rightarrow x^2 + y^2 + 6y + 9 = y^2 - 6y + 9\]
\[ \Rightarrow x^2 = - 12y\]