The Distance Formula and the Circle


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1 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation, sketch the graph of a circle In Section 10.1, we eamined the parabola. In this section, we turn our attention to another conic section, the circle. Definitions: 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. ( 2, 2 ) Definitions: The Distance Formula The distance, d, between two points ( 1, 1 ) and ( 2, 2 ) is given b d d 2( 2 1 ) 2 ( 2 1 ) 2 ( 1, 1 ) (, ) ( 2, 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 2( h) 2 ( k) 2 Squaring both sides of the equation gives the equation of the circle (h, k) r r 2 ( h) 2 ( k) 2 In general, we can write the following equation of a circle. NOTE A special case is the circle centered at the origin with radius r. Then (h, k) (0, 0), and its equation is 2 2 r 2 Definitions: Equation of a Circle The equation of a circle with center (h, k) and radius r is ( h) 2 ( k) 2 r 2 (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. 767
2 768 CHAPTER 10 GRAPHS OF CONIC SECTIONS Eample 1 3 Finding the Equation of a Circle Find the equation of a circle with center at (2, 1) and radius 3. Sketch the circle. Let (h, k) (2, 1) and r 3. Appling equation (1) ields ( 2) 2 [ ( 1)] (2, 1) ( 2) 2 ( 1) 2 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. ( 2) 2 ( 1) 2 9 CHECK YOURSELF 1 Find the equation of the circle with center at ( 2, 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). Eample 2 Finding the Center and Radius of a Circle NOTE 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) 2 ( 2) 2 9 ( 2) 2 9 ( 1) 2 ( 2) 29 ( 1) ( 1) 2 Now graph the two functions 2 29 ( 1) 2 and 2 29 ( 1) 2 on our calculator. (The displa screen ma need to be squared to obtain the shape of a circle.) Find the center and radius of the circle with equation ( 1) 2 ( 2) 2 9 Remember, the general form is ( h) 2 ( k) 2 r 2 Our equation fits this form when it is written as Note: 2 ( 2) ( 1) 2 [ ( 2)] So the center is at (1, 2), and the radius is 3. The graph is shown. 3 (1, 2) ( 1) 2 ( 2) 2 9
3 THE DISTANCE FORMULA AND THE CIRCLE SECTION CHECK YOURSELF 2 Find the center and radius of the circle with equation ( 3) 2 ( 2) 2 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. NOTE To recognize the equation as having the form of a circle, note that the coefficients of 2 and 2 are equal. NOTE The linear terms in and show a translation of the center awa from the origin. Eample 3 Finding the Center and Radius of a Circle Find the center and radius of the circle with equation Then sketch the circle. 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. First, add 1 to both sides to complete the square in. 3 ( 1, 3) ( 1) 2 ( 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) 2 ( 3) 2 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 Find the center and radius of the circle with equation Sketch the circle.
4 770 CHAPTER 10 GRAPHS OF CONIC SECTIONS CHECK YOURSELF ANSWERS 1. ( 2) 2 ( 1) center: ( 3, 2); radius 4 5 ( 2, 1) 4 ( 3, 2) 3. ( 2) 2 ( 1) 2 4; center: (2, 1); radius 2 2 (2, 1)
5 Name 10.2 Eercises Section Date In eercises 1 to 12, decide whether each equation has as its graph a line, a parabola, a circle, or none of these ( 3) 2 ( 2) ( 3) ( 3) In eercises 13 to 20, find the center and the radius for each circle ( 3) 2 ( 1) ( 3) ANSWERS In eercises 21 to 32, graph each circle b finding the center and the radius
6 ANSWERS ( 1) ( 2) ( 4) 2 ( 1) ( 3) 2 ( 2)
7 ANSWERS Describe the graph of Describe how completing the square is used in graphing circles. 35. A solar oven is constructed in the shape of a hemisphere. If the equation describes the outer edge 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 outer edge of this oven. 37. A solar water heater is constructed in the shape of a half clinder, with the water 4 suppl pipe at its center. If the water heater has a diameter of m, what is the equation that describes its outer edge? 3 773
8 ANSWERS A solar water heater is constructed in the shape of a half clinder with circumference described b the equation What is its diameter if the units for the equation are meters? 41. A circle can be graphed on a calculator b plotting the upper and lower semicircles on the same aes. For eample, to graph , we solve for : This is then graphed as two separate functions, Y and Y In eercises 39 to 42, use that technique to graph each circle ( 3) ( 5)
9 ANSWERS 42. ( 2) 2 ( 1) Each of the following equations defines a relation. Write the domain and the range of each relation. 43. ( 3) 2 ( 2) ( 1) 2 ( 5) ( 3) ( 2) Answers 1. Parabola 3. Line 5. Circle 7. Circle 9. None of these 11. Parabola 13. Center: (0, 0); radius: Center: (3, 1); radius: Center: ( 1, 0); radius: Center: (3, 4); radius: Center: (0, 0); radius: 2 Center: (0, 0); radius: Center: (0, 0); radius: Center: (0, 0); radius: ( 1) ( 4) 2 ( 1) 2 16 Center: (1, 0); radius: 3 Center: (4, 1); radius: 4 ( 1) Center: (1, 0); radius: 3 ( 4) 2 ( 1) 2 16 Center: (4, 1); radius: 4 775
10 Center: (0, 2); radius: 4 Center: (2, 1); radius: ( 2) 2 16 Center: (0, 2); radius: ( 2) 2 ( 1) 2 4 Center: (2, 1); radius: Circle with radius 0; center at (1, 2) cm 24.4 cm ( 5) ( 5) 2 36 ( 5) Domain: Domain: 5 5 Range: 2 6 Range:
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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