Question

Find the equation of the parabola if the focus is at (0, −3) and the vertex is at (−1, −3)

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 the axis and directrix.

 It is given that the vertex and the focus of a parabola are (−1, −3) and (0, −3), respectively.

Thus, the slope of the axis of the parabola is zero.

And, the slope of the directrix 

∴ \[\frac{r + 0}{2} = - 1, \frac{s - 3}{2} = - 3\]
\[ \Rightarrow r = - 2, s = - 3\] 

∴ Required equation of the directrix: 

\[x + 2 = 0\] 

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

\[x + 2 = 0\] 

Draw PM perpendicular to \[x + 2 = 0\] 

Then, we have: 

\[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 0 \right)^2 + \left( y + 3 \right)^2 = \left| \frac{x + 2}{\sqrt{1}} \right|^2 \]
\[ \Rightarrow x^2 + y^2 + 6y + 9 = x^2 + 4x + 4\]
\[ \Rightarrow y^2 + 6y - 4x + 5 = 0\]