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

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

Let be the area of an ellipse and be the area of its minor circle.

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

The first row has been done for you.

Use the table in part (a) to state a relationship between the fraction and the fraction .

Use your relationship in part (b) to find in the statement below:

If is 25 times bigger than , then the area of the ellipse is times bigger than the area of its minor circle.

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

Let be the area of an ellipse and be the area of its major circle.

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

You may use any values from the table in question 2, where relevant.

The first row has been done for you.

Use the formula

and the formula

to show using algebra that

Using the result in part (b), describe the relationship between values in the column of the table in part (a) and values in the column.

The difference in the area between an ellipse and its minor circle is shaded below.

Find a formula for the area of this shaded region.

Give your answer fully factorised.

Find a formula for the area of the region between an ellipse and its major circle.

Give your answer fully factorised.

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

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

One approximate formula is

Use this formula to estimate the circumference of an ellipse with and .

Give your answer fully simplified and in terms of .

The true circumference of an ellipse with and is , rounded to the nearest whole number.

Explain, with a reason, whether the estimate in part (a) is an overestimate or an underestimate.

Substitute into the formula

and simplify the answer.

Explain what this represents.

A second approximate formula for the circumference of an ellipse is

where is the radius of a circle that has the same area as the ellipse.

For example, an ellipse with and has an area of , which is the same area as a circle of radius , so

Use the formula to find an approximation to the circumference of the ellipse with and .

You must show your working clearly.

Give your answer fully simplified and in terms of .

The true circumference of an ellipse with and is , rounded to the nearest whole number.

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

The area of an ellipse is

The area of a circle, radius , is

Use algebra to show that the radius, , that makes both areas equal to each other is

On the diagram below, the point is called the focus of an ellipse.

The focus is:

• a point on the line

• that is a distance away from

• and that forms a right-angled triangle,

• with a hypotenuse of , as shown.

The length is not shown.

Use Pythagoras' theorem to show that the length is given by

A third formula to approximate the circumference of an ellipse is

where is the horizontal distance from the centre of the ellipse to its focus, , as found in part (a).

Use this formula to work out an estimate for the circumference of an ellipse with and .

You must show your working clearly.

Give your answer in the form where is a fully simplified fraction.

The true circumference of an ellipse with and is , rounded to 2 decimal places.

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