Question

How do you find the equation of a parabola with vertex at origin and focus (-2, 0)?

Answer 1

Since the vertex of the parabola and the focus is given to us we will have to start solving the sum by directly writing down the general equation of the parabola. Before that you need to find the axis of symmetry by using the focus given to us. After this, substitute the information given to us in the general equation and you get the desired equation.

Complete step by step solution:
In this question we have been given the vertex as origin and focus as (-2, 0).
We will start the solution by noting down the general formula for the equation of a parabola.
Since the vertex is at (0, 0) and focus is at (-2, 0) we have the axis of symmetry as y=0
The general equation of parabola when the axis of symmetry is y=0 is given by ${(y - {\text{k}})^2} = 4p\left( {x - {\text{h}}} \right)$, focus is given by $({\text{h + p, k}})$ and the directrix is given by $x = {\text{h - p}}$.
Now since the vertex given to us is (0, 0) substituting it in general formula we get
$\Rightarrow {\left( {y - 0} \right)^2} = 4p\left( {x - 0} \right) \\\ \Rightarrow {y^2} = 4px - - - \left( 1 \right) \\\$
Also from the focus (-2, 0) we get our directrix as x=2.
Now substituting the value of directrix in general equation of directrix we get the value of as -2
Now substituting all this values in equation (1) we get,
$\Rightarrow {y^2} = 4 \times \left( { - 2} \right)x \\\ \Rightarrow {y^2} = - 8x \\\ \Rightarrow {y^2} + 8x = 0 \\\$

Hence we obtain the equation of parabola with vertex at origin and focus (-2, 0) as ${y^2} + 8x = 0$.

Note: in this question the vertex of the parabola is given as (0, 0) which makes the process a little easier. Also in this solution we have used the simplified general equation of the parabola corresponding to its given axis of symmetry. You can use the basic general formula as well which will lead you to the same equation.