1CCircles


 * Circles**


 * Circles are one of the group of shapes called Conic Sections (not in course).


 * Circles can be defined geometrically (not in course).

A circle with can be defined using coordinate geometry as the relation:
 * radius = r
 * centre on the origin

... ... ** x 2 + y 2 = r 2 **


 * Domain: x Î [–r, r]
 * Range: y Î [–r, r]

Example 1


 * Sketch x 2 + y 2 = 9


 * Solution;**


 * This is a circle with
 * radius = 3
 * centre at the origin (0, 0).


 * Domain: x Î [–3, 3]
 * Range: y Î [–3, 3]


 * Vertices (–3, 0) . (+3, 0) . (0, –3) . (0, +3)


 * x-intercepts x = –3, x = 3


 * y-intercepts y = –3, y = 3

General Equation for a Circle

If the circle with a radius of r is translated we get the general equation for a circle:
 * h units to the right and
 * k units up,

... ... ** (x – h) 2 + (y – k) 2 = r 2 **


 * Domain: x Î [h – r, h + r]
 * Range: y Î [k – r, k + r]

Example 2
 * Sketch (x – 4) 2 + (y – 1) 2 = 16


 * Solution:**


 * This is a circle with a
 * radius = 4
 * centre at (4, 1).


 * Domain: x Î [0, 8]
 * Range: y Î [–3, 5]


 * Vertices : (0, 1) . (8, 1) . (4, –3) . (4, 5)


 * y-intecept : y = 1


 * x-intercepts (find by setting y = 0)

math \\ . \qquad (x - 4)^2 + (-1)^2 = 16 \\. \\ . \qquad (x – 4)^2 = 15 \\. \\ . \qquad x-4 = \pm \sqrt {15} \\. \\ . \qquad x = 4 \pm \sqrt {15} math

Example 3
 * Sketch the graph of x 2 + y 2 + 6x + 4y – 3 = 0

Aim: To rewrite this equation in the form of the general equation for a circle.


 * Group the x terms together and group the y terms together.

... ... x 2 + y 2 + 6x + 4y – 3 = 0

... ... x 2 + 6x + y 2 + 4y = 3


 * Complete the square for the x terms by adding 9 to both sides.
 * Complete the square for the y terms by adding 4 to both sides.

... ... x 2 + 6x + 9 + y 2 + 4y + 4 = 3 + 9 + 4.

... ... (x + 3) 2 + (y + 2) 2 = 16

This is a circle with
 * radius = 4,
 * translated 3 to the left
 * translated 2 down.
 * centre at (–3, –2)




 * Radius = 4
 * Centre = (–3, –2)


 * Domain: x Î [–7, 1]
 * Range: y Î [–6, 2]


 * Vertices : (–7, –2) . (1, –2) . (–3, –6) . (–3, 2)


 * y-intercepts

math . \qquad y = - 2 \pm \sqrt {7} math


 * x-intercepts

math . \qquad x = -3 \pm 2 \sqrt {3} math

Sketching Circles on a Calculator

While the calculator is set to Function mode, we need to enter equations in the form y = f(x)

Rearranging ... ... x 2 + y 2 = 9

We get math \\ . \qquad y^2=9-x^2 \\. \\ . \qquad y = \pm \sqrt {9 - x^2} math

This would have to be entered as two separate functions math \\ . \qquad y1 = +\sqrt {9 - x^2} \\. \\ . \qquad y2 = - \sqrt {9 - x^2} math


 * y1 will draw the top half of the circle,
 * y2 will draw the bottom half.


 * Due to limitations caused by the way functions are drawn on the calculator,
 * the circle will look incomplete near the x-axis
 * with the two functions not joining up properly.
 * the lines are too steep for the calculator to draw effectively

A better circle can be drawn using parametric mode and the parametric equations of a circle.

Return to Summary of Conic Sections .