Question

The line x=25/4 is directrix of an ellipse and its associated focus is the point (4,0),
find the equation of the ellipse.

Answer 1

The vertex is half way from the focus to the directrix.

Answer 2

Yes.
Then i know that one vertex is (41/8,0)
How does that help me?

Answer 3

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

Ringing any bells? You will be required on examination to be acquainted with this form.

Answer 4

ok, but that's the parabola equation.
We're talking about ellipse.

Answer 5

Right. Just saw that. Sorry. You will not be required to know about a parabola a ellipse question.

Answer 6

Directrix, x = 25/4 is "outside" the focus (4,0), so we are to construe via symmetry that the other focus is at (-4,0)? That's not entirely clear to me. I'm thinking we need one more clue. Center at the origin? Eccentricity? Something.

Answer 7

based on http://mathworld.wolfram.com/Ellipse.html the directrix will be at \(\bf \cfrac{a^2}{c} \implies \cfrac{25}{4} \implies\cfrac{a=5}{c=4}\)

then we also know that "c" is the distance from the center to the focus and that \(\bf c= \sqrt{a^2-b^2}\)

we know "c" and we know "a"
c = 4
a = 5
so what's "b"?
well \(\bf \cfrac{a^2}{c} \implies \cfrac{25}{4} \implies\cfrac{a=5}{c=4}\\
c = \sqrt{a^2-b^2} \implies b^2 = a^2-c^2 \implies b = \sqrt{a^2-c^2}\)

Answer 8

Well, that's true if the center of the ellipse is at the origin.
And the problem never say that.
So i have to assume that as a condition in order to solve the problem?

Answer 9

well, ... to be honest, we know one focal point, but we dunno where the center might be
it could be the origin, or the at (4,4) or (0,-4)

Answer 10

(4, -4) rather

Answer 11

but I'd assume any of those valid points for the center will be a valid equation for the requested ellipse

Answer 12

the picture at wolfram uses the 0,0 origin, yes, but the equation for the directrix is based on the "a" and "c" component, which aren't exclusive to the origin, they're just a scalar value

Answer 13

It is easy to assume the origin. If I wanted to impress the teacher, I might assume the eccentricity. It may appear as though one were paying attention! e = 0.8, for example.