Find an equation of the parabola with focus $\left( \frac { 15 } { 2 } , 0 \right)$ and directrix $x = - \frac { 13 } { 2 }$.
A)
$y = \frac { x ^ { 2 } } { 2 } + 14$
B) $y = 28 x ^ { 2 } - 14$
C) $y ^ { 2 } = 14 x - 28$
D)
$y = \frac { x ^ { 2 } } { 14 } + 2$
E) $y ^ { 2 } = 28 x - 14$

Question

Quizplus · Accepted Answer

Since the focus is to the right of the vertex and the directrix is vertical, the parabola opens to the right. Thus, the equation of the parabola can be written in the form $y^2=4px$, where $p$ is the distance from the vertex to the focus (and also from the vertex to the directrix). The distance from the vertex to the focus is $\frac{15}{2}$ minus the x-coordinate of the vertex, and the distance from the vertex to the directrix is the absolute value of $-\frac{13}{2}$ minus the x-coordinate of the vertex. Setting these distances equal to each other and solving for the x-coordinate of the vertex yields $\frac{1}{2}$, so the vertex is at $\left(\frac{1}{2},0ight)$. Using $p=\frac{15}{2}-\frac{1}{2}=7$ gives $y^2=28x$, which matches choice $\boxed{	extbf{(E)}}$.

Find an Equation of the Parabola with Focus (152,0)\left( \frac { 15 } { 2 } , 0 \right)(215​,0)

Find an Equation of the Parabola with Focus $\left( \frac { 15 } { 2 } , 0 \right)$