By A. C. Burdette (Auth.)
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Extra info for An Introduction to Analytic Geometry and Calculus
R, or (x - h)2 + (y- k)2 = r 2 , (3-2) the condition that P must satisfy if it is on the circle. Furthermore, if the Fig. 3-7 32 3. NONLINEAR EQUATIONS AND GRAPHS coordinates x and y of a point satisfy (3-2), it is r units distant from (A, k) and therefore is a point on the circle. Thus (3-2) is an equation of the circle with center at (//, k) and radius r. If we square the terms in the left member of (3-2) and simplify, we obtain x 2 + y2 - 2hx -2ky + d = 0, (3-3) another equation of this circle.
Hence the curve is infinite in extent in the j-direction. Solving for y, we obtain y= ± y * 2 - 4 . Since x2 — 4 is negative when — 2 < x < 2 , | we see that no points on the curve lie between the lines x = — 2 and x = 2. The results in the preceding examples could be expressed in terms of excluded values of x and y. Thus, in Example 3-6 we could say that there are no excluded values of y but values of x > 4 are excluded. Example 3-7 has |x| > 5 and | j | > 5 excluded; Example 3-8 has \x\ < 2 excluded.
Example 3-13. Discuss and graph the equation x2 + y2 — 4x + 6y + 22 = 0. We reduce this equation to the form of (3-2) by completing squares and obtain (x2 - 4x + 4) + (y2 + 6y + 9) = - 2 2 + 4 + 9 or ( x - 2 ) 2 + (>; + 3 ) 2 = - 9 . Since r2 = — 9, r is not real and this equation has no real locus. Example 3-14. Find an equation for the circle which is tangent to the line y = — 1 and whose center is at ( — 3, 2). From (3-2) we note that the equation of a circle can be written down immediately if we know the center and radius.