1. Parabolas (Conic Sections)

L3CALC

Author: Juyoung Park & Sedecember juyoung.parkk8@gmail.com & a3060100s@gmail.com

---

We have dealt with parabolas for the past however long we’ve done mathematics for. But, what is the definition of a parabola?

A definition of a parabola is a symmetrical curve where EVERY point on the curve is at an equidistance from a ‘fixed point’ to a ‘fixed straight line’.

The term ‘fixed point’, is the focus (pl. foci), and the term ‘fixed straight line’, is the directrix.

Now, let’s try to form the equation of the parabola below.

The equation can be formed by using the definition of “at every point onthe curve, is equidistant from the focus and the directrix”
Diagram link here

Now form a formula using the definition.

$$\sqrt{(x-c{)}^2+y^2}=x+c$$

LHS is the distance to the focus point, and the RHS is also the distance to the focus point. Since the definition states that they’re at equidistances, this equation is true for all parabolas with a foci at $(c,0)$.

After algebraic manipulation, we get

$$(x-c{)}^2+y^2=(x+c{)}^2$$ $$x^2-2cx+c^2+y^2=x^2+2cx+c^2$$ $$4cx=y^2$$ $$\boxed{y^2=4cx}$$

We have obtained the formula of the parabola. If you check the formula sheet yourself, then you’ll see that the formula that we obtained matches the one on the formula sheet (except the formula sheet uses $(a,0)$ as the foci!)