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A Circle is a familiar shape. If you are wondering what type of equation has a circle for a graph, this section is all about satisfying your curiosity.
Given a point C(a,b) on the Cartesian coordinate system and a nonnegative real number r, there is one and only one circle whose center is C and whose radius is r. The circle centered at C and of radius r is the set of all points on the Cartesian coordinate system that are at a distance of exactly r from C:
Equation Of A Circle: Center-Radius FormA point P(x,y) is on such a circle if and only if the distance between P and C is r. Using the distance formula, we get: . Squaring both sides, we get the equivalent equation: That is, a point P(x,y) is on the circle, centered at (a,b) and of radius r, if and only if (x,y) is a solution of the equation So that the equation of the circle is
Example 1 (i) Write the equation of the circle centered at (1,-1) and of radius 3. (ii) The equation of a circle is given by: . Find its radius and the coordinates of its center. Solution (i) Using the center-radius form, we get: , or equivalently: . (ii) The equation of the circle may be written as: . This indicates that the center has coordinates (-1,4) and radius .
Equation Of A Circle: Expanded FormIn the previous example, we found the equation of the circle centered at (1,-1) and of radius 3 to be: . If we expand the squares on the left hand side, we get: , or equivalently: . This last equation is also the equation of the same circle. It is just written differently. The advantage of the first equation is that the coordinates of the center and the radius are readable from the equation, which is not the case in the last equation.
More generally, if a circle is centered at the point C(a,b) and is of radius r, its equation is: Expanding the squares on the left side, we get: . Subtracting from both sides, we get: If we let , , and , the last equation becomes: . This is the equation of the circle in expanded form.
Example 2 A circle is given by its equation as follows: . Find its radius and the coordinates of its center.
Solution If we can write the equation of this circle in the form: , then we would conclude that the center has coordinates (a,b) and radius r. The question is therefore: How do we transform the equation to make it look like ? Recall completing the square? That's what we will use to achieve our goal. Notice that . Similarly, . So that the equation of the circle, , may now be written as: . Adding 13 to both sides, we get: , or equivalently: . This last equation says that the center is at (-3,2) and that the radius is 4.
Example 3 Consider the equation: . Is this the equation of a circle? If so, find its radius and the coordinates of its center.
Solution Completing the squares for and , the equation becomes . Adding 10 to both sides, we get: . This is the equation of a circle centered at (-1,3) and of radius 0. This circle is reduced to the point (-1,3).
Example 4 Consider the equation . Is this the equation of a circle? If so, find its radius and the coordinates of its center.
Solution Completing the squares for and , the equation becomes . Adding 10 to both sides, we get: . The left hand side of this equation is the sum of two squares and therefore can never be negative (cannot equal -5). This equation has no solutions. There are no points on its graph. The given equation is not that of a circle.
In reviewing the past three examples, the equation is not always that of a circle. Whether it is or it is not the equation of a circle depends on the values of A, B, and C. The question is therefore: under what condition(s) on A, B, and C would it be the equation of a circle? We answer that question now.
Completing the square in and in , the equation becomes: . Adding to both sides, we get: (1) Note that the left hand side of this equation is the sum of two squares and therefore is never negative. The right hand side however may be negative, zero, or positive. We discuss these three cases separately: Case 1. Suppose . In this case, equation (1) may be written as: . This is indeed in center-radius form where the center is at and the radius is .
Case
2. Suppose
In this case, equation (1) becomes:
Case 3. Suppose In this case, equation (1) has no solutions because its left hand side is never negative as the sum of two squares while its right hand side is negative. This is not the equation of a circle.
Example 5 Given the equation . For what value(s) of b is this the equation of a circle (that is not reduced to a point)?
Solution In this example, we have: A = 2, B = b, and C = -5. The given equation is that of a circle (not reduced to a point) if and only if quantity is positive (i.e., ) Solving this inequality for b, we get: b < -4 or b > 4. In conclusion, the equation is that of a circle (not reduced to a point) if and only if b is in
Example 6 A circle is centered at (1,1) and passes through the point (3,3). (i) Find the equation of such a circle in center-radius form. (ii) Find its equation in expanded form.
Solution Since the circle has center at (1,1) and passes through the point (3,3), its radius r is the distance between the points (1,1) and (3,3): . (i) The equation of the circle in center-radius form is therefore: (ii) In expanded form, its equation is:
Refer to the interactive exploration below.
You may graph a circle in two ways: (i) By entering its equation where indicated and then pushing the button "graph equation", or (ii) By choosing two points on the graph (you choose a point on the graph with a left mouse click): the first point will be the center and its distance from the second point will be the radius. In this case, the equation is also calculated and displayed.
Experiment with this functionality until you are confident about circles and their equations both is standard and expanded forms.
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