Ellipses - Example 1: Sketching 4x^2 + 9y^2 = 36

4x² + 9y² = 36 is the equation of an ellipse centred at the origin (0,0). Before we can sketch the ellipse, we need to find the vertices (i.e. the x and y intercepts) by transforming the equation to the standard form, which is:

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

where...

a is the semi-major axis
b is the semi-minor axis

So, dividing the equation by 36, we get:

\frac{x^2}{9} + \frac{y^2}{4} = 1

Or...

\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1

Thus a = 3 and b = 2, and hence the vertices are:

A = (3,0)
A' = (-3,0)
B = (0,2)
B' = (0,-2)

To fully define the ellipse, we should also find the focal points and the directrices. Thus we to find the eccentricity. We can do this through the relationship:

b^2 = a^2 - (ae)^2

The focal points (foci) are given by:

F = (ae,0)
F' = (-ae,0)

And the equations are of the directrices are:

x = \frac{a}{e}

x = -\frac{a}{e}