Find an equation for the parabola with vertex $( 5,5 )$ and directrix $y - \text { axis }$
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 $y-\text{axis}$, which is a vertical line passing through the point $(5,0)$, the focus is at the same distance vertically above the vertex. This means the focus is at $(5,10)$.
Using the formula for the distance between a point $(x,y)$ and a vertical line $x=a$, we get that the distance between a point on the parabola $(x,y)$ and the directrix $y-\text{axis}$ is $|x-5|$. So, we have:

$$\text{distance from }(x,y)\text{ to directrix} = |y-0| = |y|$$
$$\text{distance from }(x,y)\text{ to focus} = \sqrt{ (x-5)^2 + (y-10)^2 }$$

By definition of a parabola, these distances are equal. So, we have:

$$|y| = \sqrt{ (x-5)^2 + (y-10)^2 }$$
Squaring both sides and simplifying gives:

$$(y-5)^2 = 20(x-5)$$
which is in the form of a vertical parabola with vertex $(5,5)$ and directrix $y-\text{axis}$. Therefore, the answer is choice E.

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

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