Question

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Part 1
Find an equation of a parabola satisfying the given information.
Focus​ (0, −3π​), directrix y=3π

Answer 1

An equation of a parabola given a directrix being a horizontal line is x^2 = 4ax. And the directrix is y = a so a = 3pi. Therefore, the equation should be x^2 = 12pix.

Answer 2

To find the equation of a parabola given the focus and the directrix, we can use the standard form of the equation of a parabola with the vertex at the origin:

For a parabola with the focus at (0, p) and the directrix y = -p, the equation is given by:

y^2 = 4px

where:

(0, p) is the focus point.

The directrix is the horizontal line y = -p.

Given the information:

Focus (0, -3π) and directrix y = 3π

We can see that the value of 'p' is -3π (since the y-coordinate of the focus is -3π), and the directrix is y = 3π.

Now, plug the value of 'p' into the standard equation:

y^2 = 4px

y^2 = 4(-3π)x

y^2 = -12πx

So, the equation of the parabola satisfying the given information is y^2 = -12πx.