For Exercises 41–50, write the standard form of the equation of the hyperbola subject to the given conditions. (See Example 5)
Corners of the reference rectangle:
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- Exercises 45–48 give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix.arrow_forwardIn Exercises 3–10, describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve.arrow_forwardFor Exercises 13–22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) 13. 16 y? = 1 25 14. 25 y? = 1 36 y² 15. 4 = 1 36 y? 16. 9. = 1 49 17. 25y - 81x = 2025 18. 49y? 16x = 784 19. - 5x? + 7y² = -35 20. –7x + 1ly = -77 21. 25 16y 1 4x? 22. 81 16y? 1 49 225arrow_forward
- For Exercises 43–48, the equation represents a conic section (nondegenerative case). a. Identify the type of conic section. (See Example 6) b. Graph the equation on a graphing utility. 43. 4x – 4xy + 5y – 20 = 0 44. 6x + 4V3xy + 2y - 18x + 18V3y – 72 = 0 45. 2x – 6xy + 3y² - 4x + 12y – 9 = 0 46. 5x – 3xy + 2y – 6 = 0 47. 4x + 8xy + 4y – 2x – 5y – 2 = 0 48. 4x? + 8V3xy + 3y + 2x – 12y – 6 = 0arrow_forwardExercises 45–48 give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix. 45. y2 = 4x, 46. x2 = 8y, right 1, down 7 47. x2 = 6y, left 2, down 3 48. y2 = -12x, right 4, up 3 left 3, down 2arrow_forwardExercises 27–34 give equations for hyperbolas. Put each equation instandard form and find the hyperbola’s asymptotes. Then sketch thehyperbola. Include the asymptotes and foci in your sketch.27. x2 - y2 = 1 28. 9x2 - 16y2 = 14429. y2 - x2 = 8 30. y2 - x2 = 431. 8x2 - 2y2 = 16 32. y2 - 3x2 = 333. 8y2 - 2x2 = 16 34. 64x2 - 36y2 = 2304arrow_forward
- In Exercises 5–16, determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.arrow_forwardFor Exercises 27–34, an equation of a parabola x = 4py or y = 4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Examples 2-3) 27. x -4y 28. x -20y 29. 10y = 80x 30. 3y = 12x 31. 4x 40y 32. 2x 14y 33. y = 34. y = -2x = -X %3Darrow_forwardFind the equation y = ax2 + bx +c of the parabola that passes through the points (-2,0), (0, –14), (7,0) .arrow_forward
- In Exercises 17–30, find the equations of the parabolas satisfying the given conditions. The vertex of each is at the origin.arrow_forwardWhich are the foci for the hyperbola modeled by the equation (X-32 (x-1)²-1? 36 13 O (1. 10) and (1,-4) O (3.8) and (3, -8) O (1.9) and (1, -3) O (-6, 0) and (6, 0)arrow_forwardfind the equation of the parabola y = ax? + bx + c that passes through the points. (0, 3), (1, 4), (2, 3)arrow_forward
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