Parabolas

What is a parabola? How is it defined?

What is the standard parabola equation? Most people I ask would say y = x^2. Wrong!

Before I get into the mathematics of this, imagine a point and a line. Draw a locus which is equidistant (equally distant) from the point and the line, and voila you get a parabola! That point is known as the focus of the parabola, and the line, directrix. This property is often called the focus-directrix property of the parabola.


Imagine a parabola which opens to the right and has its vertex at (0,0). Let the focus of the parabola be the point (a, 0), and the directrix the equation x + a = 0. Since the distance of the line from the directrix to the parabola is equal to the distance from the focus to the parabola, then

√[(x - a)^2 - y^2] = x + a

Squaring both sides,

(x - a)^2 - y^2 = (x + a)^2

x^2 - 2ax + a^2 - y^2 = x^2 + 2ax + a^2

y^2 = 4ax --- this is the cartesian equation of the parabola.

x = at^2, y = 2at --- these are the parametric equation of the parabola.

Using implicit differentiation and simple coordinate geometry, you can work out the tangent of the parabola at any point.

An exercise for you:
Find an equation of the normal at P(4t, 4t^2) on the parabola with equation 4y = x^2. The normal meets the y-axis at the point Q. Find, in cartesian form, an equation of the locus of the point M, the mid-point of PQ, as t varies.

Loving further maths!