All points on a **circle** are the same distance from the center. The radius tells us the distance from the center to any point on the circle.

Consider the circle with a radius of 13 units shown below:

The **standard form of the equation** of a circle is

\displaystyle \left(x-h\right)^{2}+\left(y-k\right)^2=r^2

\bm{r}

radius of the circle

\bm{\left(h,k\right)}

center of the circle

\bm{\left(x,y\right)}

coordinates of any point on the circle

To check whether a point \left(x_1,y_1\right) is inside, on or outside a circle, we can compare the distance between that point and the center of the circle to the value of the radius.

Using the Pythagorean theorem, we can write these conditions as:

- If \left(x_1-h\right)^2+\left(y_1-k\right)^2<r^2 then \left( x_1,y_1 \right) is inside the circle
- If \left(x_1-h\right)^{2}+\left(y_1-k\right)^2=r^2 then \left( x_1,y_1 \right) is on the circle
- If \left(x_1-h\right)^2+\left(y_1-k\right)^2>r^2 then \left( x_1,y_1 \right) is outside the circle

Notice that these conditions are the same as substituting the point into the equation of the circle and comparing the values on each side.

Derive the equation of a circle with center \left(h,k\right) and radius r.

Worked Solution

Consider the circle shown.

a

State the coordinates of the center.

Worked Solution

b

State the radius of the circle.

Worked Solution

c

State the diameter of the circle.

Worked Solution

d

State the equation of the circle.

Worked Solution

Write the equation of the circle with the given conditions.

a

Center at \left(-1, 2 \right) and a radius of 4

Worked Solution

b

Center at \left(2, 4 \right) and a point on the circle at \left(-2, 1 \right)

Worked Solution

c

Endpoints of a diameter are \left(-1.5, 4\right) and \left(4.5,-2\right)

Worked Solution

Darnell shines a flashlight at a wall which lights up a circular region with a diameter of 4 meters. The center of the light is positioned 3 meters above the ground, and 5 meters horizontally from the left side of the wall.

a

Let the bottom left corner of the wall be the origin. Determine the equation of the circle which describes the edge of lighted area.

Worked Solution

b

Yvonne has a height of 1.66 meters and is standing against the wall, 5 meters from the left side. Determine if any part of Yvonne is in the lighted area.

Worked Solution

c

Kayoko is standing against the wall, 6 meters from the left side of the wall. Determine the greatest height that Kayoko can be without being in the lighted area. Round your answer to the nearest centimeter.

Worked Solution

Idea summary

The **standard form of the equation** of a circle is

\displaystyle \left(x-h\right)^2+\left(y-k\right)^2=r^2

\bm{r}

radius of the circle

\bm{\left(h,k\right)}

center of the circle

\bm{\left(x,y\right)}

coordinates of any point on the circle