Solve the problem.
-The parabola $x ^ { 2 } = - 24 y$ is shifted 4 units up and 6 units to the right. Find the focus and directrix of the new parabola.
A) Focus: (-2, 6); Directrix: x = 10
B) Focus: (6, -2); Directrix: y = -10
C) Focus: (6, 10); Directrix: y = 10
D) Focus: (6, -2); Directrix: y = 10

Question

Quizplus · Accepted Answer

The vertex form of the equation for a parabola is given by $(x-h)^2=4p(y-k)$, where $(h,k)$ is the vertex, $p$ is the distance between the vertex and the focus, and the directrix is the line $y=k-p$.

In this case, the original equation is $x^2=-24y$, so the vertex is at $(0,0)$ and $4p=-24$, hence $p=-6$.

After the shift, the equation becomes $(x-6)^2=-24(y-4)$. This parabola has vertex $(6,4)$. The distance between the vertex and focus is still $6$. Therefore, the focus is at $(6,4+6)=(6,10)$.

The equation of the directrix is $y=k-p=4+6-6=4$, but this is the equation of the directrix before the shift. After the shift, the directrix is $y=4-6=-2$. Therefore, the answer is choice D.

Solve the Problem x2=−24yx ^ { 2 } = - 24 yx2=−24y

Solve the Problem $x ^ { 2 } = - 24 y$