Programmed tutorial: Find the Graph and Equation of a Circle given a Diameter: Example 1

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Try to solve this problem without using a graphing calculator.
Find the equation of the circle for which the points (-6, 5) and (2, -1) are the ends of a diameter.

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First plot the points and draw the diameter on paper. Then check by clicking “Next”.

alt="Java applet graph of 2 points and a diameter." Your browser is completely ignoring the <APPLET> tag!

State the standard equation of a circle.

What is the meaning of h and k?

They are the coordinates of the center of the circle.

To evaluate, what additional data do we need to get out of the diagram?

We need the coordinates of the center and the radius.

How is the radius R related to the diameter D?

D = 2 R

How does this help in determining h and k?

The center of the circle must be halfway between the two endpoints of the diameter.

Estimate the coordinates of that point from the graph.

The point at (-2, 2) looks like it is halfway, and may be the center of the circle.

Use the definition of midpoint to find the center precisely.

The coordinates are

This confirms what we read from the graph.

Determine the values for h and k.

h = -2 and k = 2

Now let’s determine D and R. How can we find D?

We can use the definition of distance with the given points.

Do it.

What can we conclude about R?

We now have the data we need.
Write the equation for this circle.

becomes

Simplify this equation.

Add the circle to your plot. Then check by clicking “Next”.

alt="Java applet graph of a circle and its diameter." Your browser is completely ignoring the <APPLET> tag!

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