Question

A parabola can be drawn given a focus of (8, 6) and a directrix of x=4 Write the equation of the parabola in any form. Answer:

Answer 1

The equation of the parabola in vertex form is
`x-6 = (1)/(16)\cdot (y-6)^(2)`.

Explanation:

Since the directrix is of the
`x = a`, then the equation of the parabola in vertex form is of the form:

`x-h = (1)/(4\cdot p)\cdot (y-k)^(2)` (1)

Where:

`x` - Dependent variable.

`y` - Independent variable.

`p` - Least distance between focus and directrix.

`h`,
`k` - Coordinate of the vertex.

The least distance between focus and directrix is determined by Pythagorean Theorem:

`p = \sqrt{(8-4)^(2)+(6-6)^(2)}`

`p = 4`

Now, we determine the location of the vertex by the following vectorial formula:

`(h,k) = F(x,y) - (0.5\cdot p, 0)` (2)

If we know that
`p = 4` and
`F(x,y) = (8,6)`, then the location of the vertex is:

`(h,k) = (8,6) -(2, 0)`

`(h,k) = (6,6)`

And the equation of the parabola in vertex form is
`x-6 = (1)/(16)\cdot (y-6)^(2)`.