1.1. An ellipse is a shape that can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, $a$, and a vertical length, $b$, measured from its centre, as shown below.

The point $F$ on the diagram below is called the focus of an ellipse.

The focus is:

• a point on the line $OA$

• that is a distance $c$ away from $O$

• and that forms a right-angled triangle, $BOF$

• with a hypotenuse of length $a$, as shown.

The length $OA=a$ is not shown.

The eccentricity of an ellipse, $e$, is the ratio of the length $c$ to the length $a$, given by

$e=\frac{c}{a}$

By finding $c$ in terms of $a$ and $b$ from the diagram, show that the eccentricity can be written in the form

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

1.2. Two formulas for approximating the circumference of an ellipse are ${C}_{1}=2 \pi \sqrt{ab}$ and ${C}_{2}=2 \pi (\frac{a+b}{2})$.

A third formula to approximate the circumference of an ellipse is

${C}_{3}=2 \pi a(1-\frac{1}{4}{e}^{2}-\frac{3}{64}{e}^{4})$

where $e$ is the eccentricity of the ellipse, as given in part (a).

Use this formula for ${C}_{3}$ to work out an estimate for the circumference of an ellipse with $a=26$ and $b=24$.

You must show your working clearly.

Give your answer correct to 2 decimal places.

1.3. The true circumference of an ellipse with $a=26$ and $b=24$ is $157.14$, rounded to 2 decimal places.

Explain whether the estimate in part (b) is an overestimate or an underestimate.