Find the midpoint of the line segment with endpoints at the given coordinates. 1. (8, 3), ( 4, 9) SOLUTION: Substitute 8, 4, 3 and 9 for x 1
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1 Find the midpoint of the line segment with endpoints at the given coordinates. 1. (8, 3), ( 4, 9) Substitute 8, 4, 3 and 9 for x 1, x 2, y 1 and y 2 respectively in the midpoint formula. 2. Substitute for x 1, x 2, y 1 and y 2 respectively in the midpoint formula. 3. ( 10, 0), ( 2, 6) Substitute 10, 2, 0 and 6 for x 1, x 2, y 1 and y 2 respectively in the midpoint formula. Find the distance between each pair of points with the given coordinates. 4. ( 5, 8), (4, 3) Substitute 5, 4, 8 and 3 for x 1, x 2, y 1 and y 2 respectively in the distance formula. esolutions Manual - Powered by Cognero Page 1
2 5. Substitute for x 1, x 2, y 1 and y 2 respectively in the distance formula. 6. (4, 5), (4, 9) Substitute 4, 4, 5 and 9 for x 1, x 2, y 1 and y 2 respectively in the distance formula. esolutions Manual - Powered by Cognero Page 2
3 7. State whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. A = 1, B = 0, and C = 1 Because the discriminant is less than 0, B = 0 and A = C, the conic is a circle. 8. A = 4, B = 0, and C = 1 Because the discriminant is less than 0 and A C, the conic is an ellipse. esolutions Manual - Powered by Cognero Page 3
4 9. A = 4, B = 0, and C = 9 Because the discriminant is greater than 0, the conic is a hyperbola. 10. A =, B = 0, and C = 0 Because the discriminant is equal to 0, the conic is a parabola. esolutions Manual - Powered by Cognero Page 4
5 11. A = 2, B = 0, and C = 0 Because the discriminant is equal to 0, the conic is a parabola. 12. A = 16, B = 0, and C = 25 Because the discriminant is less than 0 and A C, the conic is an ellipse. esolutions Manual - Powered by Cognero Page 5
6 13. A = 1, B = 0, and C = 1 Because the discriminant is less than 0, B = 0 and A = C, the conic is a circle. 14. A = 1, B = 0, and C = 4 Because the discriminant is greater than 0, the conic is a hyperbola. esolutions Manual - Powered by Cognero Page 6
7 15. A = 1, B = 0, and C = 0 Because the discriminant is equal to 0, the conic is a parabola. 16. A = 4, B = 0, and C = 16 Because the discriminant is less than 0 and A C, the conic is an ellipse. esolutions Manual - Powered by Cognero Page 7
8 17. MULTIPLE CHOICE Which equation represents a hyperbola that has vertices at ( 3, 3) and (5, 3) and a conjugate axis of length 6 units? A B C D Since the y-coordinates of the vertices are same, the orientation should be horizontal. Therefore, option B and D may be the correct choice. Since the y-coordinate of the vertices is 3, option B is the correct answer. 18. CARPENTRY Ellis built a window frame shaped like the top half of an ellipse. The window is 40 inches tall at its highest point and 160 inches wide at the bottom. What is the height of the window 20 inches from the center of the base? The length of the major and the minor axis is 160 in. and 80 in. Therefore, a = 80 and b = 40. The equation of the ellipse is. To find the height of the window, substitute 20 for x and solve for y. The height of the window is about in. esolutions Manual - Powered by Cognero Page 8
9 19. Solve each system of equations. Substitute x 2 for y in the quadratic equation and solve for x. By zero product property: Substitute the values of x in the linear equation and find the value of y. The solutions are (6, 8) and ( 8, 6). esolutions Manual - Powered by Cognero Page 9
10 20. Solve the linear equation for y. Substitute 2 x for y in the quadratic equation and solve for x. By zero product property: Substitute the values of x in the linear equation and find the value of y. The solutions are (3, 1) and. esolutions Manual - Powered by Cognero Page 10
11 21. Substitute the values of x in either of the equations and find the values of y. The solutions are. 22. Solve each system of inequalities. esolutions Manual - Powered by Cognero Page 11
12 MULTIPLE CHOICE Which is NOT the equation of a parabola? F G H J The equation Option J is the correct answer. has both x 2 and y 2 terms. Therefore, this is not the equation of a parabola. esolutions Manual - Powered by Cognero Page 12
13 25. FORESTRY A forest ranger at an outpost in the Sam Houston National Forest and another ranger at the primary station both heard an explosion. The outpost and the primary station are 6 kilometers apart. a. If one ranger heard the explosion 6 seconds before the other, write an equation that describes all the possible locations of the explosion. Place the two ranger stations on the x-axis with the midpoint between the stations at the origin. The transverse axis is horizontal. (Hint: The speed of sound is about 0.35 kilometer per second.) b. Draw a sketch of the possible locations of the explosion. Include the ranger stations in the drawing. a. Let the ranger in the Sam Houston National Forest hears the explosion first. The ranger in the primary station hears the sound 6 seconds later than the other. That is, the explosion place is 6(0.35) = 2.1 kilometers farther from primary station than from Sam Houston National Forest. Therefore, the locus of all points that are 2.1 km closer to Sam Houston National Forest than to primary station is one branch of the hyperbola. 2a = 2.1 a = 1.05 The distance between the two stations is 6 km. 2c = 6 c = 3 The equation is. b. Graph the equation. esolutions Manual - Powered by Cognero Page 13
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