Question

A parabola can be drawn given a focus of left bracket, 6, comma, 7, right bracket(6,7) and a directrix of y, equals, 3y=3. Write the equation of the parabola in any form.

Answer 1

The equation of the parabola, derived from the given focus and directrix, is x^2 - 12x - 24y + 204 = 0.

To find the equation of a parabola given the focus (h, k) and the directrix y = mx + c, you can use the standard form of the parabolic equation:

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

where p is the distance between the focus and the directrix.

In this case, the focus is given as (6, 7), and the directrix is y = 3y - 3. To get the slope-intercept form, we can simplify the equation of the directrix:

3y = 3 implies y = 1

So, the directrix is y = 1, and p is the distance between the focus and the directrix, which is the absolute value of the difference between the y-coordinate of the focus and the y-coordinate of any point on the directrix:

p = |7 - 1| = 6

Now, substitute these values into the standard form equation:

(x - 6)^2 = 4 * 6(y - 7)

Simplify further:

(x - 6)^2 = 24(y - 7)

Expand and write in a more standard form:

x^2 - 12x + 36 = 24y - 168

Move all terms to one side:

x^2 - 12x - 24y + 204 = 0

So, the equation of the parabola is x^2 - 12x - 24y + 204 = 0.