3. The Circle

Chapter Contents

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General form of a Circle

The circle with centre (0, 0) and radius r has the equation:

x² + y² = r²

The circle with centre (h, k) and radius r has the equation:

(x − h)² + (y − k)² = r²

Example 1

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Sketch the circle x² + y² = 4.

Answer

Example 2

Sketch the circle (x − 2)² + (y − 3)² = 16

Answer

Example 3

Sketch the circle (x + 4)² + (y − 5)² = 36

Answer

The General Form of the Circle

An equation which can be written in the following form (with constants D, E, F) represents a circle:

x² + y² + Dx + Ey + F = 0

This is called the general form of the circle.

Example:

Find the centre and radius of the circle

x² + y² + 8x + 6y = 0

Sketch the circle.

Answer

Exercises

1. Find the equation of the circle with centre (3/2, -2) and radius 5/2.

Answer

2. Determine the centre and radius and then sketch the circle:

3x² + 3y² − 12x + 4 = 0

Answer

3. Find the points of intersection of the circle

x² + y² − x − 3y = 0

with the line

y = x − 1.

Answer

Perpendicular Distance from a Point to a Line

4. The Parabola

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