Programmed tutorial: Find the Graph and Equation of a Circle Tangent to a Line: Example 1

If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.

Please try to do this problem without a graphing calculator.

Find the equation and graph of the circle centered at (–2, –3)
and tangent to the line

Equation 1

First consider a general graph (not for this problem) of a circle tangent to a line. In addition, the radial line out to the tangent point is shown.
How are that radial line and the tangent line related? alt="Java applet graph of a circle, radius, and a tangent line." Your browser is completely ignoring the <APPLET> tag!

They are mutually perpendicular.

How are their slopes, m₁ and m₂ related?

How can we use these facts here?

We can find the slope of the given line, and from that we can find the slope of the radial line. Then we can find the equation of the radial line by requiring it to go through the given center of the circle. Finally, we can find the coordinates of the point of tangency and get the radius of the circle from the distance between these points..

To get started, we need to plot the given data: the center point and the given line.
To plot the line, we can use the slope-intercept equation. Rewrite the equation in the slope-intercept form.

From

, we isolate y to get
Slope intercept form of equation of the tangent line

Equation 2

What can we conclude about the slope of the tangent line?

Will the line slope up to the right, or down to the right?

It will slope upward to the right.

Where will it cross the y-axis?

The y-intercept is

Plot the line and the center of the circle on paper. Then check your work by clicking “Next”.

alt="Java applet graph of the tangent line and the center of the circle." Your browser is completely ignoring the <APPLET> tag!

Add the radius to the point of tangency to your diagram. Check by clicking “Next”.

alt="Java applet graph of the tangent line and a perpendicular radius." Your browser is completely ignoring the <APPLET> tag!

What is the slope of the radial line?

Since it is perpendicular to the given line, its slope will be the negative reciprocal.

Determine the value of the slope.

Write the equation of the radial line in terms of m and an unknown intercept, b.

What additional condition must the radial line satisfy?

It must go through the center at (–2, –3).

Set up the equation expressing this.

becomes

Solve for b.

Combine these results to get the equation for the radial line.

The equation of the radial line is

Equation 3

This line and the given line intersect at the tangent point (x_T, y_T).
How can we use this information?

Those coordinates must satisfy the equations of each line.

Use Equations 2 and 3 to set this up.

, or

What is the Least Common Denominator here?

The LCD is 12.

Multiply both sides of the equation by 12.

, or with cancelling we get

, or

Solve.

We get -25x_T = 125, or
x_T = –5

Use this result to find y_T.

We can use either Equation 1 or Equation 2 or Equation 3.
Let’s use Equation 1:

, or

, or
y_T = 1

Express of the point of tangency as (x, y).

It is the point (–5, 1)

Add this point to your plot. Then use “Next” to check.

alt="Java applet graph of the tangent line and a perpendicular radius." Your browser is completely ignoring the <APPLET> tag!

We have the center of the circle and the point of tangency. What else do we need for the equation of the circle.

The radius.

How can we find that?

We can use the distance between the point of tangency and the center of the circle.

Set that up.

Substitute and solve.

Use these results to determine the equation of the circle.

The general equation

becomes

, or finally

Add the circle to your plot. Then check with the “Next” button.

alt="Java applet graph of a circle tangent to a line." Your browser is completely ignoring the <APPLET> tag!

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor