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

Exam code: 0607

2 hours22 questions

2 marks

This investigation looks at the geometry and properties of an ellipse, including its area and its circumference.

An ellipse 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.

When $a = b$ , the ellipse is a circle with radius $a$ .

Three shapes: a horizontal ellipse with a greater width than height labelled 'a > b,' a circle labeled 'a = b,' and a vertical ellipse labelled 'a < b.'

This investigation will only consider ellipses in the form $a \geq b$ .

Sketch an ellipse with $a = 5$ and $b = 2$ .

Label the two lengths on your diagram.

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2 marks

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

$A = π a b$

Complete the table below, leaving all answers in terms of $π$ .

The first example has been done for you:

$A = π \times 5 \times 2 = 10 π$

$a$	$b$	$A$
$5$	$2$	$10 π$
$8$	$3$
$9$	$1$
$4$	$4$
	$3$	$36 π$
$10$		$50 π$

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1 mark

Substitute $b = a$ into the formula

$A = π a b$

and simplify the answer.

Explain what this represents.

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3 marks

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.

A horizontal ellipse with width from centre a and height from centre b, with a > b.

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

$A = π a b$

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

Diagram of an ellipse with a dashed inner circle, labelled "minor circle."

Let $A_{E}$ be the area of an ellipse and $A_{C}$ be the area of its minor circle.

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

The first row has been done for you.

$a$	$b$	$\frac{a}{b}$	$A_{E}$	$A_{C}$	$\frac{A_{E}}{A_{C}}$
$8$	$2$	$4$	$16 π$	$4 π$	$4$
$10$	$5$
$12$	$4$
$20$	$1$

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1 mark

Use the table in part (a) to state a relationship between the fraction $\frac{a}{b}$ and the fraction $\frac{A_{E}}{A_{C}}$ .

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1 mark

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

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

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3 marks

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

$A = π a b$

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

Diagram showing an ellipse inside a dashed major circle, labelled "major circle."

Let $A_{E}$ be the area of an ellipse and $A_{D}$ be the area of its major circle.

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

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

The first row has been done for you.

$a$	$b$	$\frac{a}{b}$	$A_{E}$	$A_{D}$	$\frac{A_{E}}{A_{D}}$
$8$	$2$	$4$	$16 π$	$64 π$	$\frac{1}{4}$
$10$	$5$
$12$	$4$
$20$	$1$

How did you do?

2 marks

Use the formula

$A_{E} = π a b$

and the formula

$A_{D} = π a^{2}$

to show using algebra that

$\frac{A_{E}}{A_{D}} = \frac{b}{a}$

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1 mark

Using the result in part (b), describe the relationship between values in the $\frac{a}{b}$ column of the table in part (a) and values in the $\frac{A_{E}}{A_{D}}$ column.

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