Equation of ellipse from directrix and focus

[hatelove]

Find the equation of an ellipse with a directrix y = -2 and a focus at the origin.

I'm trying to find the polar equation first, and I learned this today but I forgot a lot of it and we're not allowed to take notes in class (professor says it helps to learn better) so I'm trying to look it up online but it's not much help because I can't find any elementary lessons on polar equations.

Anyway, I think the formula is

$$r = \frac{pe}{1 \pm sin\theta }$$

because of the position of the directrix, I know the ellipse is elongated along the y-axis. For this particular position given in the problem, I know it's a negative sign in the denominator (for the plus-minus symbol).

I know 'e' is eccentricity and to find that I need the center point but I forgot how to get it. I also need the vertex but I forgot how to find that too. But I know the lower vertex is between that given focus and directrix.

So far I have:

$$r = \frac{(2)e}{1 - sin\theta }$$

First step: how do I find the length of the major axes or a vertex?

 

[CaptainBlack]

I'm trying to find the polar equation first, and I learned this today but I forgot a lot of it and we're not allowed to take notes in class (professor says it helps to learn better) so I'm trying to look it up online but it's not much help because I can't find any elementary lessons on polar equations.

Anyway, I think the formula is

$$r = \frac{pe}{1 \pm sin\theta }$$

because of the position of the directrix, I know the ellipse is elongated along the y-axis. For this particular position given in the problem, I know it's a negative sign in the denominator (for the plus-minus symbol).

I know 'e' is eccentricity and to find that I need the center point but I forgot how to get it. I also need the vertex but I forgot how to find that too. But I know the lower vertex is between that given focus and directrix.

So far I have:

$$r = \frac{(2)e}{1 - sin\theta }$$

First step: how do I find the length of the major axes or a vertex?

The directrix/forus definition of an ellipse is the locus of points such that the ratio of the distance from the focus to the distance from the directrx is a constant less than one.

Here the focus is the origin so the x-y co-ordinates of a general point on the ellipse is \( (r \cos(\theta), r \sin(\theta))\)m so the distance of a point on the ellipse from the focus is \(d_f=r\). The distance of the point from the directrix at \(y=-2\) is \(d_d=r\cos(\theta)+2\).

So the condition for this to be an ellipse is:

\[\frac{d_f}{d_d}=\frac{r}{r\cos(\theta)+2}=e\]

rearranging:

\[r=\frac{2e}{1-e\cos(\theta))}\]

The ends of the major axis correspond to \(\theta=0\) and \(\theta=\pi/2\) ,...

CB