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Lesson 155: General Equation of a Circle
Until now we've been working with straight lines and parabolas. In this lesson, we'll learn how to write the equation of a circle that is drawn on the coordinate plane.
A circle drawn on the coordinate plane has a special equation that must be memorized. A circle with its center at the origin (0, 0), and radius r has the equation x2 + y2 = r2. We'll know if a given point is on the circle if the coordinates of that point satisfy the equation.
For example, is the point (3, 4) located on a circle of radius 5 that is centered at the origin? Our equation is x2 + y2 = r2. We'll substitute our values and see if the equation checks. We have 32 + 42 = 52. 9 + 16 = 25. 25 = 25. It checks. That point is on the circle.
What if the circle is not centered at the origin, its center is at a point (h, k)? The equation that we use is (x-h)2 + (y-k)2 = r2. Be sure to memorize this version of the equation.
Let's try it. What is the equation of a circle with radius
7, with its center at (-3, 6). Substituting, we have (x - -3)2
+ (y - 6)2 = 72. Simplifying we have
(x + 3)2 + (y - 6)2 = 49, and that is how we
will leave our equation. If we wanted to know if a particular
point lied in this circle, we could just substitute its x and y
coordinates, and see if the equation checks.
There are a wide variety of problems that involve this formula. In some cases, you might not be given the center or the radius of the circle explicitly, and you might need to do some computations to figure it out. As always, read the problem very carefully, and take it slowly and step by step. Write down and/or graph everything that you know, and then use the appropriate formulas and procedures to determine the information necessary to use the formula for the equation of a circle.
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