1001 10.2 Ellipses Here g is the acceleration due to gravity, and x and y are the horizontal and vertical distances in feet, respectively. The value of g will vary depending on the mass and size of the planet. (a) For Earth g = 32.2, while for Mars g = 12.6. Find the two equations, and graph on the same screen of a graphing calculator the paths of the two balls thrown on Earth and Mars. Use the window 30, 1804 by 30, 1004. (Hint: If possible, set the calculator mode to simultaneous.) (b) Determine the difference in the horizontal distances traveled by the two balls to the nearest foot. 57. (Modeling) Path of a Projectile (Refer to Exercise 56.) Suppose the two balls are now thrown upward at a 60° angle on Mars and the moon. If their initial velocity is 60 mph, then their paths in feet can be modeled by the following equation. y = 23x - g 3872 x2 (a) Graph on the same coordinate axes the paths of the balls if g = 5.2 for the moon. Use the window 30, 15004 by 30, 10004. (b) Determine the maximum height of each ball to the nearest foot. 58. Prove that the parabola with focus 1p, 02 and directrix x = -p has the equation y2 = 4px. Relating Concepts For individual or collaborative investigation (Exercises 59–62) (Modeling) Given three noncollinear points, we can find an equation of the form x = ay2 + by + c of the horizontal parabola joining them by solving a system of equations. Work Exercises 59–62 in order, to find the equation of the horizontal parabola containing the points 1-5, 12, 1-14, -22, and 1-10, 22. 59. Write three equations in a, b, and c, by substituting the given values of x and y into the equation x = ay2 + by + c. 60. Solve the system of three equations determined in Exercise 59. 61. Does the horizontal parabola open to the left or to the right? Why? 62. Write the equation of the horizontal parabola. Equations and Graphs of Ellipses Like the circle and the parabola, the ellipse is defined as a set of points in a plane. 10.2 Ellipses ■ Equations and Graphs of Ellipses ■ Translated Ellipses ■ Eccentricity ■ Applications of Ellipses Ellipse An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is constant. Each fixed point is a focus (plural, foci) of the ellipse.
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