Find an equation for the parabola with vertex $( 5,5 )$ and directrix $y - a x i s$
A) $( y - 5 ) ^ { 2 } = 25 x$
B) $( y - 1 ) ^ { 2 } = 5 ( x - 5 )$
C) $( x + 5 ) ^ { 2 } = 20 ( y + 5 )$
D) $y ^ { 2 } = 20 x$
E) $( y - 5 ) ^ { 2 } = 20 ( x - 5 )$

Question

Quizplus · Accepted Answer

Since the directrix is the line "y-axis", the parabola opens to the right. Thus, the equation must involve $(x-5)$ to represent the horizontal distance from the vertex. Also, since the focus is the same distance from the vertex as the directrix, and the vertex is halfway between the focus and directrix, we know that the distance from the vertex to the focus is also $5$. This gives us the equation $(y-5)^2=4p(x-5)$, where $p$ is the distance from the vertex to the focus/directrix. Solving for $p$, we have $p=5/4$. Substituting this into the equation, we get $(y-5)^2=20(x-5)$, which matches answer choice E.

Find an Equation for the Parabola with Vertex (5,5)( 5,5 )(5,5)

Find an Equation for the Parabola with Vertex $( 5,5 )$