Learning Resources

Lesson

In the last lesson we dealt with the standard form of a circle, ( x - h)² + (y - k)² = r². From it we can immediately ascertain the centre and radius of the circle and we can use this information to graph it fairly quickly. However, quite often the equation of a circle is not given in standard form and we have to do some manipulation of it in order to get it into standard form.

To understand how to do this, first consider the expansion and rearrangement of the following equation which is already in standard form:  (x - 5)² + (y + 2)² = 9

        (x -5)² + (y + 2)² = 9
        x² -10x + 25 + y² + 4y + 4 = 9   (squaring the binomial terms)
        x² + y² -10x + 4y + 25 + 4 = 9   (rearranging terms)
        x² + y² -10x + 4y + 20 = 0         (transposing & combining constant terms)

It is the last step, x² + y² -10x + 4y + 20 = 0, that we quite often start with and may be asked to give the coordinates of the centre, find the radius, and draw its graph. To do this we have to reverse the steps shown above. The best way of understanding how to do this is to work through some examples.

Example 1

Write the following equation of a circle in standard form and give the coordinates of its centre and the length of its radius: x² + y² - 4x + 10y - 20 = 0

Solution

Remember, we are reversing the expansion shown in the example above. This process is presented in a step by step approach with some commentary. To see the solution click here.

Example 2

Write the following equation of a circle in standard form and give the coordinates of its centre and the length of its radius: 6x² + 6y² + 24x - 36y - 72 = 0

Solution

Again the solution is presented in a step by step approach with some commentary. Simply click the arrows to direct the flow of the solution.

In both of the above examples, the form of the original equation is called the general form. It may be written as: Ax² + Ay² + Dx + Ey + F = 0.  Pay particular attention to the fact that the coefficients of  x² and y² are the same. This is the identifying characteristic of the equation of a circle. If they are not the same the resulting graph will not be a circle. We will explore this in a later lesson.

You now have two forms for the equation of a circle. They are:

                 Standard Form:  ( x - h)² + (y - k)² = r²

                       General Form:    Ax² + Ay² + Dx + Ey + F = 0

Learn to recognize both and be able to convert from one form to the other. If you can do this you should be able to do the work assigned for this lesson.

Activity

1. Complete Focus H on pages 257 & 258 in your text (Hint: For the second example, you may find it better to first divide through by 5 as we did in the example in this lesson, rather than factoring it out as done in your text.)
2. Complete the Focus Questions 21 & 22 on page 258.
3. Do the CYU Questions 23 - 30, 32, 33 on pages 259 & 260.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

1. Give the equation in general form of a circle with centre at (-4 , 3) and radius 5.
2. Sketch the graph of 2x² + 2y² - 20x + 8y + 50 = 0.

Solution