# Finding the Intercepts of a Circle (center at the origin)

An x-intercept is where the graph touches or crosses the x-axis.

A y-intercept is where the graph touches of crosses the y-axis.

To find an x-intercept: Let y=0 and solve for x.

To find an y-intercept: Let x=0 and solve for y.

Example:  Find the intercepts of the circle for the given equation.

Solution:

To find an x-intercept, let y=0 and solve for x.

This equation has two x-intercepts. and

To find a y-intercept, let x=0 and solve for y.

This equation has two y-intercepts. and

# Write the Equation of a Circle Given the center and the radius

Example:  Write the equation of a circle in standard form given the center of the circle is (5,7) and the radius of the circle is 6.  Then write the equation in general (expanded) form.

Solution:

The standard form of an equation of a circle is where r is the radius and   is the center.

The center of our circle is .  So and .

The radius of our circle is 6 so .

Replace h, k and r in standard form of an equation of a circle.

Simplify.

The equation of a circle in standard for with center (5,7) and a radius of 6 is .

To write the equation in general form we can start with the standard form we just found and multiply each binomial.

The equation of a circle in general form with center (5,7) and a radius of 6 is .

# Find the center and the radius from the equation

A circle is the collection of points that are equidistant to a center point.  The distance is the radius denoted r.  The center is denoted .

The standard form of an equation of a circle is where r is the radius and   is the center.

Example:  Find the center and the radius from the given equation.

Solution:  If you line up the standard form of a circle with the equation given you can determine the center and the radius.

In the binomial with x, the number after the minus sign is h.  Thus, .  In the binomial with y, the number after the minus sign is k.  Thus, .  The lines up with 16.  So which means .

This is the equation of a circle with a center of and a radius of .

Example:  Find the center and the radius from the given equation.

Solution:  Notice that one of the binomials has a + instead of the – that is in the standard form of the circle.  Rewrite the addition as subtraction.

is the same as .

The rewritten equation is as follows.

Then, if you line up the standard form of a circle with the equation given you can determine the center and the radius.

In the binomial with x, the number after the minus sign is h.  Thus, .  In the binomial with y, the number after the minus sign is k.  Thus, .  The lines up with 25.  So which means .

This is the equation of a circle with a center of and a radius of .

Example:  Find the center and the radius from the given equation.

Solution:  Notice that one of the binomials has a + instead of the – that is in the standard form of the circle.  Rewrite the addition as subtraction.

is the same as .

The other binomial doesn’t have any number added or subtracted.  We can rewrite this by subtracting zero.

is the same as .

The rewritten equation is as follows.

Then, if you line up the standard form of a circle with the equation given you can determine the center and the radius.

In the binomial with x, the number after the minus sign is h.  Thus, .  In the binomial with y, the number after the minus sign is k.  Thus, .  The lines up with 25.  So which means .

This is the equation of a circle with a center of and a radius of .