Problem involving conics. Need to find points of intersection given information about a conic.

Question

A conic has eccentricity $e=0.7$, a focus $(5,−3)$ and directrix $y=2x−7$. Find the points of intersection of the conic with line $y=−3$.

I'm really stuck on this, and have no idea even where to start.

Any help?

Answer 1

First figure out what kind of conic it is. Since the eccentricity of a parabola or circle is 1.0 (parabola) or 0.0 (circle), it's not one of those. It's either an ellipse or hyperbola, and since the eccentricity is less than 1.0, it's not a hyperbola. So it's an ellipse.

Next use the focus and directrix to figure out the equation of the ellipse, given that the general equation of an ellipse with major axis $a$ , minor axis $b$ is

$$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1$$

and the eccentricity

$$e= \sqrt{1-\dfrac{b^2}{a^2}} $$

and the focal parameter (distance from focus to directrix) is given by

$$p=\dfrac{b^2}{\sqrt{a^2-b^2}}$$

Once you have the equation of the ellipse, you should be able to figure out where it intersects with $$y= -3$$

I'll leave that to you as an exercise.