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Circles Solutions

A circle is a conic section that is called a horizontal slice .
A circle is the set of all points in a plane that are equidistant from a fixed point, the center.
A circle is named using a capital letter at its center.
The radius is the fixed distance at the center to any point on the circle.

The diameter of a circle is the line segment (chord) that passes through the center of the circle (point A) and whose endpoints (points B and C) are on the circle.

The diameter is twice the length of the radius.

Note

A circle is a special type of ellipse, which you will learn about in a lesson later in this unit.

The standard form of the equation of a circle, which has a center $(h, k)$ and a radius r, is ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ .
Points that form the diameter of a circle can be determined by adding or subtracting the value of the radius to the center.
The endpoints of the horizontal diameter are $(h + r, k)$ and $(h - r, k)$ and determine the domain of the circle.
The endpoints of the vertical diameter are $(h, k + r)$ and $(h, k - r)$ and determine the range of the circle.

Because you can determine h, k, and the radius r from the equation, you do not need to write them down unless specifically directed. However, it may be helpful, especially when working with negative values of h and k.

Example 1

Write the equation of circles A, B, and C in standard form.

Plan

Identify the center and radius

Write the equation

Implement

Circle A

$(- 5, 4), r = 4$

${(x - (- 5))}^{2} + {(y - 4)}^{2} = 4^{2} {(x + 5)}^{2} + {(y - 4)}^{2} = 16$

Circle B

$(0, 0), r = 3$

${(x - 0)}^{2} + {(y - 0)}^{2} = 3^{2} x^{2} + y^{2} = 9$

Circle C

$(3, - 2), r = 6$

${(x - 3)}^{2} + {(y - (- 2))}^{2} = 6^{2} {(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Note

If you can, simplify the equation in one step!

Example 2

Write the equation of each circle in the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ with a radius of 1.5 units and a center of $(7.9, - 24)$ .

${(x - 7.9)}^{2} + {(y - (- 24))}^{2} = 1 . 5^{2} {(x - 7.9)}^{2} + {(y + 24)}^{2} = 2.25$

Remember not all values for the center and radius of a circle are integers.