Solve the problem.
-The parabola $x ^ { 2 } = - 32 y$ is shifted 4 units up and 7 units to the left. Find the focus and directrix of the new parabola.
A) Focus: (-7, 12); Directrix: y = 12
B) Focus: (-7, -4); Directrix: y = 12
C) Focus: (7, -12); Directrix: y = -12
D) Focus: (-7, -4); Directrix: y = -4

Question

Quizplus · Accepted Answer

The equation for the parabola after shifting it 4 units up and 7 units to the left is:
$$(x+7)^2 = -32(y+4)$$
Comparing this with the standard form of the equation of a parabola:
$$(x-h)^2 = 4p(y-k)$$
where (h,k) is the vertex and 4p is the distance between the vertex and the focus/directrix, we can see that the vertex of the new parabola is (-7,-4).

We also know that the parabola opens downwards, since the coefficient of y is negative.

Using the formula for the distance between the vertex and the focus/directrix:
$$4p = -32$$
$$p = -8$$

Since the parabola opens downwards, the focus will be 8 units below the vertex, and the directrix will be 8 units above the vertex. Therefore, the focus is at (-7,-4-8) = (-7,-12) and the directrix is the horizontal line passing through (-7,-4+8) = (-7,4), which is y = 12. Hence, the answer is B.

Solve the Problem x2=−32yx ^ { 2 } = - 32 yx2=−32y

Solve the Problem $x ^ { 2 } = - 32 y$