48 CHAPTER 1 Graphs 150. If two distinct lines have the same slope but different x -intercepts, can they have the same y -intercept? 151. If two distinct lines have the same y -intercept but different slopes, can they have the same x -intercept? 152. Which form of the equation of a line do you prefer to use? Justify your position with an example that shows that your choice is better than another. Have reasons. 153. What Went Wrong? A student is asked to find the slope of the line joining ( ) −3, 2 and ( ) − 1, 4 . The student states that the slope is 3 2 . Is the student correct? If not, what went wrong? 145. Carpentry Carpenters use the term pitch to describe the steepness of staircases and roofs. How is pitch related to slope? Investigate typical pitches used for stairs and for roofs. Write a brief essay on your findings. 146. Can the equation of every line be written in slope-intercept form? Why? 147. Does every line have exactly one x -intercept and one y -intercept? Are there any lines that have no intercepts? 148. What can you say about two lines that have equal slopes and equal y -intercepts? 149. What can you say about two lines with the same x -intercept and the same y -intercept? Assume that the x -intercept is not 0. 1.6 Circles Now Work the ‘Are You Prepared?’ problems on page 52. • Completing the Square (Section A.3, p. A29) • Square Root Method (Section A.6, p. A49) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Write the Standard Form of the Equation of a Circle (p. 48) 2 Graph a Circle by Hand and by Using a Graphing Utility (p. 49) 3 Work with the General Form of the Equation of a Circle (p. 51) 1 Write the Standard Form of the Equation of a Circle One advantage of a coordinate system is that it enables us to translate a geometric statement into an algebraic statement, and vice versa. Consider, for example, the following geometric statement that defines a circle. Figure 71 shows the graph of a circle. To find the equation, let ( ) x y , represent the coordinates of any point on a circle with radius r and center ( ) h k , . Then the distance between the points ( ) x y , and ( ) h k , must always equal r.That is, by the distance formula, ( ) ( ) − + − = x h y k r 2 2 or, equivalently, ( ) ( ) − + − = x h y k r 2 2 2 DEFINITION Circle A circle is a set of points in the xy -plane that are a fixed distance r from a fixed point ( ) h k , . The fixed distance r is called the radius , and the fixed point ( ) h k , is called the center of the circle. Figure 71 ( ) ( ) − + − = x h y k r 2 2 2 y (x, y) (h, k) r x DEFINITION Standard Form of an Equation of a Circle The standard form of an equation of a circle with radius r and center ( ) h k , is ( ) ( ) − + − = x h y k r 2 2 2 (1) Need to Review? The distance formula is discussed in Section 1.2, pp. 13–16.
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