Question

A parabola, with its vertex at the origin, has a directrix at y = 3. Which statements about the parabola are true? Select two options.
The focus is located at (0,–3).
The parabola opens to the left.
The p value can be determined by computing 4(3).
The parabola can be represented by the equation x2 = –12y.
The parabola can be represented by the equation y2 = 12x.

Answer 1

Answer:

Step-by-step explanation:

Let's answer these questions in an all-encompassing kind of explanation. If you plot the vertex and the directrix, you see that the vertex is below the directrix. Because of the fact that a parabola opens AWAY from the directrix, and wraps itself around the focus, we know it's an upside down parabola of the form

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

The p value from the equation is a distance, specifically the distance between either the vertex and the directrix, or the vertex and the focus. The vertex is exactly in the middle of the directrix and the focus. So that tells us that the focus is 3 units below the vertex (because the directrix is 3 units above the vertex). We also know from this that p = 3.

Filling in the equation with a vertex of (0, 0) which is our h and k respectively:

$4(3)(y-0)=-(x-0)^2$ which simplifies to

$12y=-x^2$ and multiplying both sides by -1:

$-12y=x^2$. This is not standard form, but it matches what your equation is in the choices. So to sum up:

The focus is located at (0, -3) and the first choice is true.

The parabola opens upside down and the second choice is not true.

The p value is found by counting the units between the vertex and the directrix, so the third choice is not true.

We solved the equation by filling in the values for h, k, and p and got that the equation in the fourth choice is true.

So the fifth choice is not true.

Answer 2

Using concepts from the equation of a parabola, it is found that:

• The focus is located at (0,–3).

• The parabola can be represented by the equation x2 = –12y.

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Equation of a parabola:

The equation of a parabola is given by:

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

• The vertex is (h,k).
• The focus is at (h,k+p).
• The directrix is y = k - p.

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• Vertex at the origin means that $h = 0, k = 0$.

• Directrix at y = 3 means that $p = -3$.
• Directrix at the y-axis means the parabola opens upwards.
• Thus, the focus is: $(0, 0 - 3) = (0,-3)$
• The p-value is: 4(-3) = -12.
• The equation of the parabola is:

$x^2 = -12y$

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The correct options are:

• The focus is located at (0,–3).

• The parabola can be represented by the equation x2 = –12y.

A similar problem is given at https://brainly.com/question/15546249