Write an equation for the parabola with vertex at the origin.
-Through $( 6 , - 6 \sqrt { 6 } )$, symmetric with respect to the $x$-axis
A) $y ^ { 2 } = 36 x$
B) $x = \sqrt { 6 } y ^ { 2 }$
C) $x ^ { 2 } = - \sqrt { 6 } y$
D) $x = 36 y ^ { 2 }$

Question

Quizplus · Accepted Answer

The given parabola has its vertex at the origin and is symmetric with respect to the x-axis, indicating it opens either to the right or left. The point (6, -6√6) lies on the parabola, suggesting it opens to the right (positive x-direction) since x is positive for the given point. The general form for such a parabola is $y^2 = 4ax$, where $a$ is the distance from the vertex to the focus. Substituting the given point into the equation, $y^2 = (-6\sqrt{6})^2 = 36 \cdot 6 = 216$, and $x = 6$, we find $216 = 4a \cdot 6$. Solving for $a$ gives $a = 9$, so the equation becomes $y^2 = 4 \cdot 9 \cdot x = 36x$, which matches option A.

Write an Equation for the Parabola with Vertex at the Origin