Systems of Quadratic Equations Problem: Figure 12-7g shows an ellipse, a hyperbola, and

Chapter 12, Problem C.2

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Systems of Quadratic Equations Problem: Figure 12-7g shows an ellipse, a hyperbola, and a line. The equations for each pair of these graphs form a system of equations. In this problem you will solve the systems by finding the points at which these graphs intersect each other. a. Solve the ellipse and hyperbola system graphically, to one decimal place. b. Solve the ellipse and line system graphically, to one decimal place. c. Solve the hyperbola and line system graphically, to one decimal place. Figure 12-7g d. The equations of the ellipse and hyperbola in Figure 12-7g are x2 y2 + 4x 5 = 0 x2 + 4y2 36 = 0 Quick! Tell which is which. How do you know? e. Solve the ellipse and hyperbola system algebraically. To do this, first eliminate y by adding a multiple of the first equation to the second. Solve the resulting quadratic equation for x. Finally, substitute the two resulting x-values into one of the original equations and calculate y. What do you notice about two of the four solutions? How does this observation agree with the graphs? f. Solve the ellipse and hyperbola system numerically. Do this by plotting the two graphs on your grapher and then using the intersect feature. Do the answers agree with parts a and e? g. The equation of the line in Figure 12-7g is 2x y = 5 Solve the ellipseline system algebraically. To do this, first solve the linear equation for y in terms of x and then substitute the result for y in the ellipse equation. After you solve the resulting quadratic equation, substitute the two values of x into the linear equation to find y. Do the answers agree with part b? h. Solve the hyperbolaline system algebraically. Which solution does not appear in the graphical solution of part c?