Investigations (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

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 formula for the area of an ellipse, $A$, is given by

$A=\mathit{\pi }ab$

A minor circle refers to the biggest circle that fits inside an ellipse, with the same centre, as shown.

A major circle refers to smallest circle that fits outside an ellipse, with the same centre, as shown.

The following notation will be used for the different areas:

• $A_E$ is the area of an ellipse

• $A_C$ is the area of its minor circle

• $A_D$ is the area of its major circle

Complete the table below, leaving answers fully simplified and in terms of $\pi$ (where necessary).

Parts of the table have been done for you.

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 formula for the area of an ellipse, $A$, is given by

$A=\mathit{\pi }ab$

A minor circle refers to the biggest circle that fits inside an ellipse, with the same centre, as shown.

A major circle refers to smallest circle that fits outside an ellipse, with the same centre, as shown.

The following notation is used for the different areas:

• $A_E$ is the area of an ellipse

• $A_C$ is the area of its minor circle

• $A_D$ is the area of its major circle

Use the area formulas for $A_E$, $A_C$ and $A_D$ to prove algebraically that

$\frac{A_E}{A_C}$

is the reciprocal of

$\frac{A_E}{A_D}$

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 formula for the area of an ellipse, $A$, is given by

$A=\mathit{\pi }ab$

In the diagram below, the area of the circle, radius $R$, is equal to the area of the ellipse, $\pi ab$.

Use algebra to show that

$R=\sqrt{ab}$

There is no algebraic formula for the circumference of an ellipse, $C$.

However, there are many formulas that give approximations to the circumference of an ellipse in the form

$2\pi r$

One approximate formula uses $r=\sqrt{ab}$ from part (a), giving:

$C_1=2\pi \sqrt{ab}$

A second approximate formula uses $r$ as the mean of $a$ and $b$, giving:

$C_2=2\pi \left(\frac{a+b}{2}\right)$

The table below compares the approximations $C_1$ and $C_2$ to the true circumferences, $C$.

Complete the table, giving answers correct to 1 decimal place.

The first two rows have been done for you.

Explain, with evidence from the table in part (b), which approximation out of $C_1$ and $C_2$ you would not recommend for ellipses that are very flat.

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

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

Two formulas for approximating the circumference of an ellipse are $C_1=2\pi \sqrt{ab}$ and $C_2=2\pi \left(\frac{a+b}{2}\right)$.

A third formula to approximate the circumference of an ellipse is

$C_3=2\pi a\left(1-\frac{1}{4}{e}^2-\frac{3}{64}{e}^4\right)$

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.

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.

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.

An ellipse can be rotated 360$°$ about the $x$-axis to form a 3D shape called a prolate spheroid, as shown, where $a>b$.

The total surface area of a prolate spheroid, $A$, is given by the formula

$A=2\pi a^2\left[1-e^2+\frac{\sqrt{1-e^2}}{e}\left(\frac{\theta \pi }{180}\right)\right]$

where $e=\frac{\sqrt{a^2-b^2}}{a}$ is the eccentricity of the ellipse, and $\theta$ is the acute angle in degrees that satisfies the equation

$\sin \theta =e$

Use the information above to find the total surface area of a prolate spheroid with $a=2$ and $b=\sqrt{3}$.

Leave your answer in the exact form

$A=p\pi +q\sqrt{3}{\pi }^2$

where $p$ is an integer and $q$ is a fraction, both of which you should find.