How to find directrix of conic

Hello I am having some confusion with conic sections.

For example, I am asked to find the directrix and eccentricity of the ellipse given by the formula

$$\frac{x^2}{9}+\frac{y^2}{16}=1$$

So, what I know is that this ellipse has vertices $(0,4)$ and $(0,-4)$

I also know it has points at $(-3,0)$ and $(3,0)$ as the solve the equation

I know that the foci are at $c=(0,\sqrt{7})$ and $-c=(0,-\sqrt{7})$

Now , if we let $\epsilon$=eccentricity then we would have $\frac{PF}{PD}=\epsilon$

the distance from the point $(-3,0)$ to $(0,\sqrt{7})$ is $4$,

But I do not know how to find the directrix,

I am looking for any help to understand, thanks

Since the major axis of this ellipse is vertical, the foci have coordinates (0,be), (0,-be), and directrices have equation y=b/e, y=-b/e. b is the length of the semi-major axis which is 4.

Since x = 0 is the equation major axis i.e. (b>a) the formula for directrices is
$$y=+b/\epsilon$$ and $$y=-b/\epsilon$$ according to the equation $$x^2/3^2 + y^2/4^2 = 1$$ b = 4 and $$\epsilon = \sqrt (1-a^2/b^2)=\sqrt 7/4$$

• Thanks. Can you eleaborate? For example, what formula are you using for $\epsilon$ ? Commented Oct 11, 2015 at 19:05
• @Quality Please visit this site Formulae Commented Oct 12, 2015 at 7:05

It is next only a small step next, proceeding from the definition of eccentricity.

EDIT1: Consider the points only on y-axis $O,F,P,D$ bottom to top.

$$PF=OP-OF=4-c= 4-\sqrt7$$

$$\epsilon < 1 = \frac{c}{a}= \frac{\sqrt{7}}{4}$$ $$\frac{PD}{PF}= \frac{1}{ \epsilon }$$

The above holds good for any point on the ellipse.From this particularly,the distance from end of major axis to the horizontal directrix $PD$ can be found out:

$$PD= \frac{4(4 \sqrt{7} -7)}{7}.$$

• Thank you, how do you get that $$PF=4-\sqrt{7}$$, because isnt the x distance from the vertex (-3,0) to the focus is 3, and the height is sqrt(7) so shouldnt the distance be d^2=3^2+7 ie d=4? Commented Oct 11, 2015 at 19:20
• If above edit is not clear I would upload a picture. Forget (-3,0), the directrix can be horizontal also as it is in this case. Commented Oct 12, 2015 at 5:58