# How do you write the equation of the parabola in vertex form given vertex at (10, 0) and a directrix x = -2?

May 31, 2017

${y}^{2} = 48 \left(x - 10\right)$

#### Explanation:

As the vertex is $\left(10 , 0\right)$ and directrix is $x = - 2$, the distance of vertex from directrix is $10 + 2 = 12$ and as vertex is to the right of directrix, focus will be further to the right of vertex by a distance of $12$ units. Hence, focus is $\left(22 , 0\right)$.

Now parabola is the locus of a point $\left(x , y\right)$, whose distance from focus, which is $\left(22 , 0\right)$ and directrix $x + 2 = 0$ is always same.

As distance from directrix is |x+2} and from focus is $\sqrt{{\left(x - 22\right)}^{2} + {y}^{2}}$. As such equation of parabola is

${\left(x - 22\right)}^{2} + {y}^{2} = {\left(x + 2\right)}^{2}$

or ${x}^{2} - 44 x + 484 + {y}^{2} = {x}^{2} + 4 x + 4$

or ${y}^{2} = 48 x - 480 = 48 \left(x - 10\right)$

graph{(y^2-48x+480)(x+2)((x-10)^2+y^2-0.2)((x-22)^2+y^2-0.2)=0 [-30.33, 49.67, -18.08, 21.92]}

May 31, 2017

${y}^{2} = 48 \left(x - 10\right)$

#### Explanation:

Look at the diagram -

The curve opens to the right, hence its equation is -

${\left(y - k\right)}^{2} = 4 \times a \times \left(x - h\right)$

Where-

$h = 10$ - x-coordinate of the vertex
$k = 0$ - y-coordinate of the vertex.
$a = 12$ - distance from the vertex to focus.

${\left(y - 0\right)}^{2} = 4 \times 12 \times \left(x - 10\right)$

${y}^{2} = 48 \left(x - 10\right)$