Find the equation of the parabola with vertex at $( 7,5 )$ and which passes through $( 6,7 )$ and $( 6,3 )$.
A) $( y - 5 ) ^ { 2 } = - 8 ( x - 7 )$
B) $\quad ( y - 3 ) ^ { 2 } = 2 ( x - 6 )$
C) $( y - 5 ) ^ { 2 } = - 4 ( x - 7 )$
D) $( y + 5 ) ^ { 2 } = 4 ( x + 7 )$
E) $( y - 5 ) ^ { 2 } = - 4 ( x + 7 )$

Question

Quizplus · Accepted Answer

We know that the vertex form of the equation of a parabola is:

$$(y-k)^2=4a(x-h)$$

where (h,k) is the vertex. So, substituting the given vertex (7,5) we get:

$$(y-5)^2=4a(x-7)$$

Now, the parabola passes through (6,7) and (6,3), so we can substitute these two points into the equation:

$$(7-5)^2=4a(6-7) \quad 	ext{and} \quad (3-5)^2=4a(6-7)$$

Simplifying these equations, we get:

$$4a=-4 \quad 	ext{and} \quad 4a=4$$

The only value that satisfies both equations is $a=-1$. Substituting this back into the equation, we get:

$$(y-5)^2=-4(x-7)$$

which matches choice (C). Therefore, the answer is C.

Find the Equation of the Parabola with Vertex At (7,5)( 7,5 )(7,5)

Find the Equation of the Parabola with Vertex At $( 7,5 )$