Question

A parabola can be drawn given a focus of (-6,3) and a directrix of x = -4. Write

the equation of the parabola in any form.

Answer 1

The equation of the parabola is 4x - 4y = -25.

Step-by-step explanation:

The equation of a parabola can be written in the form y = ax + bx², where a and b are constants.

To find the equation of the parabola given a focus of (-6,3) and a directrix of x = -4, we can use the formula for the distance from a point to a line.

The distance between the focus and a point on the parabola is equal to the distance between that point and the directrix. Using the distance formula, we can set up an equation to solve for a:

d = |x - (-6)| + sqrt((y - 3)²) = |x + 6| + sqrt(y - 3)²

d = |x + 6| + sqrt(y - 3)² = |x + 4|

Next, we can square both sides of the equation to eliminate the absolute value:

(x + 6)² + (y - 3)² = (x + 4)²

Expanding and simplifying:

x² + 12x + 36 + y² - 6y + 9 = x² + 8x + 16

Combining like terms:

4x - 4y = -25

So the equation of the parabola is 4x - 4y = -25.

Answer 2

`(y-3)^2=-4(x+5)`

Step-by-step explanation:

Given the directrix is vertical, the parabola must have a horizontal axis since it's perpendicular to the directrix.

Therefore, we can use the form
`(y-k)^2=4p(x-h)` where the focus is
`(h+p,k)` and the directrix is
`x=h-p`.

Therefore:

`h+p=-6`

`h-p=-4`

`2h=-10`

`h=-5`

So:

`x=h-p`

`-4=-5-p`

`1=-p`

`-1=p`

This means that the parabola opens to the left since
`p<0`. Remember that
`p` describes the distance between the vertex and focus point.

We can tell that
`k=3` given the focus point

In conclusion, the equation of the parabola is
`(y-3)^2=-4(x+5)`