Question

What is the standard equation for a parabola with focus (3,5) and directrix x=7

Answer 1

x = -8(y - 5)² + 3

Explanation:

When info about focus and directrix is given, we want to use the standard equation of a parabola x - h = 4p(y - k)^2.

p is the signed distance between focus and vertex (or that between directrix and vertex). Here the directrix, x = 7, lies to the right of the vertex, (3, 5), which indicates that this horizontal parabola opens to the left. The total vertex to directrix distance is 7 - 3, or 4; dividing this result by 2 results in 2; thus p = -2 (a signed distance).

Now we substitute the given and calculated info into the equation

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

x - 3 = 4(-2)(y - 5)², or

x - 3 = -8(y - 5)², or x = -8(y - 5)² + 3

Answer 2

`(y-5)^2=8(x-5)`

Explanation:

We are given that the focus of a parabola is (3, 5) and the directrix is x = 7.

The y-coordinate of the vertex should be same as the focus (k = 5). So the x-coordinate of the vertex would be:

p + (3) = 7 - p

2p = 7 - 3

p = 4/4

p = 2

The x-coordinate of the vertex would be:

h= p + (3)

h = 2 + 3

h = 5

The vertex coordinate would be: (h, k) = (5, 5)

For a vertex (h, k), the formula for equation would be

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

`(y-5)^2=4 * 2(x-5)`

`(y-5)^2=8(x-5)`