Question

Find the standard form of the equation of the parabola with a focus at (3, 0) and a directrix at x = -3.

Answer 1

Answer on Question #45102 – Math – Analytic Geometry

Question. Find the standard form of the equation of the parabola with a focus at $(3,0)$ and a directrix at $x=-3$.

Solution. Recall that a standard form of the equation of parabola is

$y^{2}=2px,$

where $p>0$. In this case the directrix of parabola is given by the equation

$x=-p/2,$

and the focus has coordinates:

$F(p/2,0).$

In our case we have the following two identities:

- of directrix:

$x=-3=-p/2$

- the focus:

$(p/2,0)=(3,0).$

It follows from each of them that

$p/2=3\qquad\Rightarrow\qquad p=6.$

Therefore such a parabola with a focus at $(3,0)$ and a directrix at $x=-3$ exists and the standard form of its equation is

$y^{2}=2\cdot 6\,x=12x.$

Answer. $y^{2}=12x$.