Find the equation of the parabola with vertex at $( 5,6 )$ and which passes through $( 4,8 )$ and $( 4,4 )$.
A) $\quad ( y - 6 ) ^ { 2 } = - 8 ( x - 5 )$
B) $( y - 4 ) ^ { 2 } = 2 ( x - 4 )$
C) $( y - 6 ) ^ { 2 } = - 4 ( x + 5 )$
D) $( y + 6 ) ^ { 2 } = 4 ( x + 5 )$
E) $( y - 6 ) ^ { 2 } = - 4 ( x - 5 )$

Question

Quizplus · Accepted Answer

Since the vertex of the parabola is given as $(5,6)$, we can assume that the equation of the parabola is of the form $(y - 6)^2 = 4p(x - 5)$, where $p$ is the distance between the vertex and the focus. Now we can use the fact that the parabola passes through $(4,8)$ and $(4,4)$:

$(8 - 6)^2 = 4p(4 - 5)$ and $(4 - 6)^2 = 4p(4 - 5)$

Simplifying these equations, we get:

$4p = -4$ and $4p = 4$

Only the second equation is consistent with the given information, which means that $p = 1$.

Substituting this value of $p$ in the equation above, we get:

$(y - 6)^2 = -4(x - 5)$

Simplifying further gives:

$(y - 6)^2 = -4x + 20$

Therefore, the correct answer is E.

Find the Equation of the Parabola with Vertex At (5,6)( 5,6 )(5,6)

Find the Equation of the Parabola with Vertex At $( 5,6 )$