Conic Sections - Why do I need to know all these terms (foci, latus rectum, directrix, etc)? When will I use them?

Question

I believe in learning something because I want to. If I do not want to learn about a subject or concept, I will not learn it well and master it. I am currently learning about conic sections, and I am wondering why so many terms are being thrown at me. I am often faced with problems like:

Find the foci and the eccentricity of the hyperbola $\dfrac{(x-2)^2}{9}-(y-3)^2=25$

I pretty much always think, "Yeah, yeah, the foci and stuff like that. When will I actually use these? I mean, if I will not actually use this, why am I learning about this and doing these problems?"

There must be some use for this invisible line called the directrix. There must be some use for the foci (or whatever that is, I learnt that it is just some kind of special point(s) related to the conic sections). But I am never told what the applications of these terms are. Can someone please tell me why I have to know about this; maybe it will help me appreciate conics more. Thanks.

Answer 1

Three fundamental applications of conics:

1) Kepler's laws of planetary motion (Elipse)

2) Contstruction of parabolic reflector (Parabola)

3) Mirror constructions (Hyperbola)

They all use the foci.

Answer 2

One practical, direct example of this stuff I can think of immediately is the following property of parabolas: if $P$ is a point on a parabola, then the line from $P$ to the focus and the line through P parallel to the axis meet the tangent line to P in the same angle.

In particular, if you make a mirror in the shape of a paraboloid, and put a light bulb at the focus, the reflected light will all be emitted in roughly the same direction: i.e. you get a flashlight.

The opposite direction is useful for telescopes: the light from distant objects comes in parallel to the axis, reflects off of a parabolic mirror, and is gathered together at the focus, where you can affix another mirror to reflect the light to the viewing apparatus.