ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
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1 SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as a line, a parabola, or a circle. Write the equation of a circle in standard form and graph the circle a b c d e f 0 but we will develop the standard form for a circle through the definition of a circle. A circle is the set of all points in the plane equidistant from a fied point, called the center of the circle. The distance between the center of the circle and an point on the circle is called the radius of the circle. The distance formula is central to an discussion of conic sections. (, ) The Distance Formula The distance d between two points ( 1, 1 ) and (, ) is given b d d ( 1 ) ( 1 ) ( 1, 1 ) (, 1 ) We can use the distance formula to derive the algebraic equation of a circle, given its center and its radius. Suppose a circle has its center at a point with coordinates (h, k) and radius r. If (, ) represents an point on the circle, then, b its definition, the distance from (h, k) to (, ) is r. Appling the distance formula, we have r ( h ) ( k ) Squaring both sides of the equation gives the equation of the circle (h, k) r (, ) r ( h) ( k) In general, we can write the following equation of a circle. 785
2 786 Chapter 1 Conic Sections A special case is the circle centered at the origin with radius r. Then (h, k) (0, 0), and its equation is r Equation of a Circle The equation of a circle with center (h, k) and radius r is ( h) ( k) r (1) Equation (1) can be used in two was. Given the center and radius of the circle, we can write its equation; or given its equation, we can find the center and radius of a circle. E a m p l e 1 3 (, 1) ( ) ( 1) 9 Finding the Equation of a Circle Find the equation of a circle with center at (, 1) and radius 3. Sketch the circle. Let (h, k) (, 1) and r 3. Appling equation (1) ields ( ) [ ( 1)] 3 ( ) ( 1) 9 To sketch the circle, we locate the center of the circle. Then we determine four points 3 units to the right and left and up and down from the center of the circle. Drawing a smooth curve through those four points completes the graph. CHECK YOURSELF 1 Find the equation of the circle with center at (, 1) and radius 5. Sketch the circle. Now, given an equation for a circle, we can also find the radius and center and then sketch the circle. We start with an equation in the special form of equation (1). E a m p l e Finding the Center and Radius of a Circle ( 1) ( ) 9 Remember, the general form is ( h) ( k) r
3 Our equation fits this form when it is written as Section 1. The Circle 787 The circle can be graphed on the calculator b solving for, then graphing both the upper half and lower half of the circle. In this case, ( 1) ( ) 9 ( ) 9 ( 1) ( ) 9 ( 1 ) 9 ( 1 ) Now graph the two functions 9 ( 1 ) and 9 ( 1 ) on our calculator. (The displa screen ma need to be squared to obtain the shape of a circle.) ( 1) [ ( )] 3 Note: ( ) So the center is at (1, ), and the radius is 3. The graph is shown. CHECK YOURSELF 3 (1, ) ( 1) ( ) 9 ( 3) ( ) 16 Sketch the circle. To graph the equation of a circle that is not in standard form, we complete the square. Let s see how completing the square can be used in graphing the equation of a circle. E a m p l e 3 To recognize the equation as having the form of a circle, note that the coefficients of and are equal. The linear terms in and show a translation of the center awa from the origin. Finding the Center and Radius of a Circle Then sketch the circle. 6 1 We could, of course, simpl substitute values of and tr to find the corresponding values for. A much better approach is to rewrite the original equation so that it matches the standard form.
4 788 Chapter 1 Conic Sections First, add 1 to both sides to complete the square in. ( 1) ( 3) 9 3 ( 1, 3) Then add 9 to both sides to complete the square in We can factor the two trinomials on the left (the are both perfect squares) and simplif on the right. ( 1) ( 3) 9 The equation is now in standard form, and we can see that the center is at ( 1, 3) and the radius is 3. The sketch of the circle is shown. Note the translation of the center to ( 1, 3). CHECK YOURSELF 3 Sketch the circle. 4 1 CHECK YOURSELF ANSWERS 1. ( ) ( 1) 5.. ( 3) ( ) (, 1) 4 ( 3, ) 3. ( ) ( 1) 4. (, 1)
5 E e r c i s e s Parabola. Circle 3. Line 4. Line 5. Circle 6. Parabola 7. Circle 8. Line 9. None of These 10. Circle 11. Parabola 1. None of These 13. Center: (0, 0); radius: Center: (0, 0); radius: Center: (3, 1); radius: Center: ( 3, 0); radius: Center: ( 1, 0); radius: Center: (0, 3); radius: Center: (3, 4); radius: Center: 5, 3 ; radius: Center (0, 0); radius:. Center (0, 0); radius: 5 3. Center (0, 0); radius: 3 4. Center (0, 0); radius: 4 5. Center (1, 0); radius: 3 6. Center (0, ); radius: 4 7. Center (4, 1); radius: 4 8. Center ( 3, ); radius: 5 In Eercises 1 to 1, decide whether each equation has as its graph a line, a parabola, a circle, or none of these ( 3) ( ) ( 3) ( 3) In Eercises 13 to 0, find the center and the radius for each circle ( 3) ( 1) ( 3) In Eercises 1 to 3, graph each circle b finding the center and the radius ( 1) 9 6. ( ) ( 4) ( 1) ( 3) ( ) 5 789
6 790 Chapter 1 Conic Sections 9. Center (0, ); radius: Center (3, 0); radius: Center (, 1); radius: 3. Center (1, 3); radius: Describe the graph of Describe how completing the square is used in graphing circles. 33. Circle with radius zero cm;.4 cm m ( 3 ) 9 ( 3 ) ( 5 ) 3 6 ( 5 ) ( ) 1 5 ( ) 35. A solar oven is constructed in the shape of a hemisphere. If the equation 500 describes the circumference of the oven in centimeters, what is its radius? 36. A solar oven in the shape of a hemisphere is to have a diameter of 80 cm. Write the equation that describes the circumference of this oven. 37. A solar water heater is constructed in the shape of a half clinder, with the water suppl pipe at its center. If the water heater has a diameter of 4 3 m, what is the equation that describes its circumference? 38. A solar water heater is constructed in the shape of a half clinder having a circumference described b the equation What is its diameter if the units for the equation are meters? A circle can be graphed on a calculator b plotting the upper and lower semicircles on the same aes. For eample, to graph 16, we solve for : Domain: 7 1; range: Domain: 4; range: Domain: 5 5; range: Domain: 8 4; range: 6 6 This is then graphed as two separate functions, 1 6 and 1 6 In Eercises 39 to 4, use that technique to graph each circle ( 3) ( 5) ( ) ( 1) 5 Each of the following equations defines a relation. Write the domain and the range of each relation. 43. ( 3) ( ) ( 1) ( 5) ( 3) ( ) 36
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