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 find 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
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
To perform this procedure in reverse, recall how to complete the square. If you have forgotten this procedure click here for examples and explanation.
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 Go forward one frame in the global buttons and 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.
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
Focus Questions p.258. Complete 21 and 22
Think About p.258 (2 items)
Check Your Understanding p.259 - 260. Complete 23 to 34 inclusive
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.
After you do the assigned activities, continue on to the Test Yourself section below for a quick quiz on this lesson.
Test Yourself
- Give the equation in general form of a circle with centre at (-4 , 3) and radius 5.
- Sketch the graph of 2x2 + 2y2 - 20x + 8y + 50 = 0.
Click here for suggested solution.