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<< 1.3. Distance and Midpoint Formulas

1.4. Circles

In Section 1.2 we gave two formulas for the equation of a line in the xy-plane. How about a circle? To find such an equation, recall that a circle is defined as the set of points in a plane equidistant from a given point. The simplest point to start with is the origin, so we first derive the equation of a circle centered at the origin. Later we derive the general equation of a circle in the xy-plane.

Suppose we wish to find the equation of the circle C centered about the origin with radius R, where R is a positive constant. We know that every point P=(x,y) on C satisfies the equation d(O,P) = R. Using formula (1.3.1), we see that d(O,P) = √x^{2}+y^{2}, so the equation of C is √x^{2}+y^{2} = R, or more simply

**(1.4.1)**x^{2}+ y^{2}= R^{2}.

What about circles which are not centered at the origin? Suppose the center of the circle is the point Q = (a,b). Every point P=(x,y) on the circle of radius R with center Q must be a distance R from Q, i.e. we have d(Q,P) = R. Once again, using (1.3.1), we see that this amounts to the formula

**(1.4.2)**(x-a)^{2}+ (y-b)^{2}= R^{2}.

So how does one graph a circle? The easiest circles to graph are those centered at the origin. The idea is to find as many points on the circle as possible and join them. Consider the circle of radius 5 centered about the origin. The equation of C is x^{2} + y^{2 }= 25. Solving for y in terms of x, we find y = ±√25 - x^{2}. Using this formula, for a given value of x from -5 to 5, we can compute the two values of y corresponding to points (x, y) on C. Plugging in x = -5, we find y = ±√25 - 5^{2} =±√25 - 25^{} = 0, so that the point (-5,0) lies on C. Similarly, we see that (5,0) lies on C. Plugging in x = -4, we find y = ±√25 - 4^{2} = ±√25 - 16^{} = ±√9 = 3, so that the points (-4,3) and (-4,-3) also lie on C. It is straightforward to show in the same way that the points (-3,4), (-3,-4), (0,5), (0,-5), (3,4), (3,-4), (4,3), and (4,-3). We have thus found twelve points on C. Graphing these points, it is fairly easy to graph C by joining them with a smooth curve. The graph of C along with these points is shown below. Of course, it is also possible and probably advisable to compute a few more points.

Figure 1.4.1: Graph of a Circle

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