1028 CHAPTER 10 Analytic Geometry (c) In 4x2 - 16x + 9y2 + 54y = -61, the coefficients of the x2- and y2-terms are unequal and both positive, so the equation might represent an ellipse but not a circle. (It might also represent a single point or no points at all.) 4x2 - 16x + 9y2 + 54y = -61 Given equation 41x2 - 4x + 91y2 + 6y 2 = -61 Factor out 4, and factor out 9. 41x2 - 4x + 4 - 42 + 91y2 + 6y + 9 - 92 = -61 Complete the square. 41x2 - 4x + 42 - 16 + 91y2 + 6y + 92 - 81 = -61 Distributive property 41x - 222 + 91y + 322 = 36 Factor. Add 16 and 81 on each side. 1x - 222 9 + 1y + 322 4 = 1 Divide by 36. This equation represents an ellipse having center 12, -32. See Figure 38. (d) Since only one variable in x2 - 6x + 8y - 7 = 0 is squared (x, and not y), the equation represents a parabola with a vertical axis of symmetry. Isolate the term with y (the variable that is not squared) on one side. x2 - 6x + 8y - 7 = 0 Given equation 8y = -x2 + 6x + 7 Isolate the y-term. 8y = -1x2 - 6x 2 + 7 Regroup terms, and factor out -1. 8y = -1x2 - 6x + 9 - 92 + 7 Complete the square. 8y = -1x2 - 6x + 92 + 9 + 7 Distributive property; -1-92 = +9 8y = -1x - 322 + 16 Factor, and add. y = - 1 8 1x - 322 + 2 Multiply by 1 8 . y - 2 = - 1 8 1x - 322 Subtract 2. The parabola has vertex 13, 22 and opens down, as shown in Figure 39. An equivalent form for this parabola is 1x - 322 = -81y - 22. S Now Try Exercises 33, 35, 37, and 41. x y 4x2 – 16x + 9y2 + 54y = –61 (2, –3) –3 –1 2 5 0 Figure 38 Multiply 41-42 = -16 and 91-92 = -81. Multiply each term by 1 8 . Here, 1 8 1162 = 2. (3, 2) x2 – 6x + 8y – 7 = 0 y 3 2 6 –2 –1 x 0 Figure 39 EXAMPLE 2 Determining Type of Conic Section Identify and sketch the graph of 4y2 - 16y - 9x2 + 18x = -43. SOLUTION 4y2 - 16y - 9x2 + 18x = -43 Given equation 41y2 - 4y - 91x2 - 2x 2 = -43 Factor out 4, and factor out -9. 41y2 - 4y + 4 - 42 - 91x2 - 2x + 1 - 12 = -43 Complete the square. 41y2 - 4y + 42 - 16 - 91x2 - 2x + 12 + 9 = -43 Distributive property 41y - 222 - 91x - 122 = -36 Factor. Add 16, and subtract 9. Be careful with this step.
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