Review Exercises

Distance, Midpoint, and the Parabola

Calculate the distance and midpoint between the given two points.

1. $\left(0,2\right)$ and $\left(-4,-1\right)$

2. $\left(6,0\right)$ and $\left(-2,-6\right)$

3. $\left(-2,4\right)$ and $\left(-6,-8\right)$

4. $\left(\frac{1}{2},-1\right)$ and $\left(\frac{5}{2},-\frac{1}{2}\right)$

5. $\left(0,-3\sqrt{2}\right)$ and $\left(\sqrt{5},-4\sqrt{2}\right)$

6. $\left(-5\sqrt{3},\sqrt{6}\right)$ and $\left(-3\sqrt{3},\sqrt{6}\right)$

Determine the area of a circle whose diameter is defined by the given two points.

7. $\left(-3,3\right)$ and $\left(3,-3\right)$

8. $\left(-2,-9\right)$ and $\left(-10,-15\right)$

9. $\left(\frac{2}{3},-\frac{1}{2}\right)$ and $\left(-\frac{1}{3},\frac{3}{2}\right)$

10. $\left(2\sqrt{5},-2\sqrt{2}\right)$ and $\left(0,-4\sqrt{2}\right)$

Rewrite in standard form and give the vertex.

11. $y=x^2-10x+33$

12. $y=2x^2-4x-1$

13. $y=x^2-3x-1$

14. $y=-x^2-x-2$

15. $x=y^2+10y+10$

16. $x=3y^2+12y+7$

17. $x=-y^2+8y-3$

18. $x=5y^2-5y+2$

Rewrite in standard form and graph. Be sure to find the vertex and all intercepts.

19. $y=x^2-20x+75$

20. $y=-x^2-10x+75$

21. $y=-2x^2-12x-24$

22. $y=4x^2+4x+6$

23. $x=y^2-10y+16$

24. $x=-y^2+4y+12$

25. $x=-4y^2+12y$

26. $x=9y^2+18y+12$

27. $x=-4y^2+4y+2$

28. $x=-y^2-5y+2$

Circles

Determine the center and radius given the equation of a circle in standard form.

29. ${\left(x-6\right)}^2+y^2=9$

30. ${\left(x+8\right)}^2+{\left(y-10\right)}^2=1$

31. $x^2+y^2=5$

32. ${\left(x-\frac{3}{8}\right)}^2+{\left(y+\frac{5}{2}\right)}^2=\frac{1}{2}$

Determine standard form for the equation of the circle:

33. Center $\left(-7,2\right)$ with radius $r=10.$

34. Center $\left(\frac{1}{3},-1\right)$ with radius $r=\frac{2}{3}.$

35. Center $\left(0,-5\right)$ with radius $r=2\sqrt{7}.$

36. Center $\left(1,0\right)$ with radius $r=\frac{5\sqrt{3}}{2}.$

37. Circle whose diameter is defined by $\left(-4,10\right)$ and $\left(-2,8\right).$

38. Circle whose diameter is defined by $\left(3,-6\right)$ and $\left(0,-4\right).$

Find the x- and y-intercepts.

39. ${\left(x-3\right)}^2+{\left(y+5\right)}^2=16$

40. ${\left(x+5\right)}^2+{\left(y-1\right)}^2=4$

41. $x^2+{\left(y-2\right)}^2=20$

42. ${\left(x-3\right)}^2+{\left(y+3\right)}^2=8$

43. $x^2+y^2-12y+27=0$

44. $x^2+y^2-4x+2y+1=0$

Graph.

45. ${\left(x+8\right)}^2+{\left(y-6\right)}^2=4$

46. ${\left(x-20\right)}^2+{\left(y+\frac{15}{2}\right)}^2=\frac{225}{4}$

47. $x^2+y^2=24$

48. ${\left(x-1\right)}^2+y^2=\frac{1}{4}$

49. $x^2+{\left(y-7\right)}^2=27$

50. ${\left(x+1\right)}^2+{\left(y-1\right)}^2=2$

Rewrite in standard form and graph.

51. $x^2+y^2-6x+4y-3=0$

52. $x^2+y^2+8x-10y+16=0$

53. $2x^2+2y^2-2x-6y-3=0$

54. $4x^2+4y^2+8y+1=0$

55. $x^2+y^2-5x+y-\frac{1}{2}=0$

56. $x^2+y^2+12x-8y=0$

Ellipses

Given the equation of an ellipse in standard form, determine its center, orientation, major radius, and minor radius.

57. $\frac{{\left(x+12\right)}^2}{16}+\frac{{\left(y-10\right)}^2}{4}=1$

58. $\frac{{\left(x+3\right)}^2}{3}+\frac{y^2}{25}=1$

59. $x^2+\frac{{\left(y-5\right)}^2}{12}=1$

60. $\frac{{\left(x-8\right)}^2}{5}+\frac{\left(y+8\right)}{18}=1$

Determine the standard form for the equation of the ellipse given the following information.

61. Center $\left(0,-4\right)$ with $a=3$ and $b=4.$

62. Center $\left(3,8\right)$ with $a=1$ and $b=\sqrt{7}.$

63. Center $\left(0,0\right)$ with $a=5$ and $b=\sqrt{2}.$

64. Center $\left(-10,-30\right)$ with $a=10$ and $b=1.$

Find the x- and y-intercepts.

65. $\frac{{\left(x+2\right)}^2}{4}+\frac{y^2}{9}=1$

66. $\frac{{\left(x-1\right)}^2}{2}+\frac{{\left(y+1\right)}^2}{3}=1$

67. $5x^2+2y^2=20$

68. $5{\left(x-3\right)}^2+6y^2=120$

Graph.

69. $\frac{{\left(x-10\right)}^2}{25}+\frac{{\left(y+5\right)}^2}{4}=1$

70. $\frac{{\left(x+6\right)}^2}{9}+\frac{{\left(y-8\right)}^2}{36}=1$

71. $\frac{{\left(x-\frac{3}{2}\right)}^2}{4}+{\left(y-\frac{7}{2}\right)}^2=1$

72. ${\left(x-\frac{2}{3}\right)}^2+\frac{y^2}{4}=1$

73. $\frac{x^2}{2}+\frac{y^2}{5}=1$

74. $\frac{{\left(x+2\right)}^2}{8}+\frac{{\left(y-3\right)}^2}{12}=1$

Rewrite in standard form and graph.

75. $4x^2+9y^2-8x+90y+193=0$

76. $9x^2+4y^2+108x-80y+580=0$

77. $x^2+9y^2+6x+108y+324=0$

78. $25x^2+y^2-350x-8y+\mathrm{1,216}=0$

79. $8x^2+12y^2-16x-36y-13=0$

80. $10x^2+2y^2-50x+14y+7=0$

Hyperbolas

Given the equation of a hyperbola in standard form, determine its center, which way the graph opens, and the vertices.

81. $\frac{{\left(x-10\right)}^2}{4}-\frac{{\left(y+5\right)}^2}{16}=1$

82. $\frac{{\left(x+7\right)}^2}{2}-\frac{{\left(y-8\right)}^2}{8}=1$

83. $\frac{{\left(y-20\right)}^2}{3}-{\left(x-15\right)}^2=1$

84. $3y^2-12{\left(x-1\right)}^2=36$

Determine the standard form for the equation of the hyperbola.

85. Center $\left(-25,10\right)$, $a=3$, $b=\sqrt{5}$, opens up and down.

86. Center $\left(9,-12\right)$, $a=5\sqrt{3}$, $b=7$, opens left and right.

87. Center $\left(-4,0\right)$, $a=1$, $b=6$, opens left and right.

88. Center $\left(-2,-3\right)$, $a=10\sqrt{2}$, $b=2\sqrt{3}$, opens up and down.

Find the x- and y-intercepts.

89. $\frac{{\left(x-1\right)}^2}{4}-\frac{{\left(y+3\right)}^2}{9}=1$

90. $\frac{{\left(x+4\right)}^2}{8}-\frac{{\left(y-2\right)}^2}{12}=1$

91. $4{\left(y-2\right)}^2-x^2=16$

92. $6{\left(y+1\right)}^2-3{\left(x-1\right)}^2=18$

Graph.

93. $\frac{{\left(x-10\right)}^2}{25}-\frac{{\left(y+5\right)}^2}{100}=1$

94. $\frac{{\left(x-4\right)}^2}{4}-\frac{{\left(y-8\right)}^2}{16}=1$

95. $\frac{{\left(y-3\right)}^2}{9}-\frac{{\left(x-6\right)}^2}{81}=1$

96. $\frac{{\left(y+1\right)}^2}{4}-\frac{{\left(x+1\right)}^2}{25}=1$

97. $\frac{y^2}{27}-\frac{{\left(x-3\right)}^2}{9}=1$

98. $\frac{x^2}{2}-\frac{y^2}{3}=1$

Rewrite in standard form and graph.

99. $4x^2-9y^2-8x-90y-257=0$

100. $9x^2-y^2-108x+16y+224=0$

101. $25y^2-2x^2-100y+50=0$

102. $3y^2-x^2-2x-10=0$

103. $8y^2-12x^2+24y-12x-33=0$

104. $4y^2-4x^2-16y-28x-37=0$

Identify the conic sections and rewrite in standard form.

105. $x^2+y^2-2x-8y+16=0$

106. $x^2+2y^2+4x-24y+74=0$

107. $x^2-y^2-6x-4y+3=0$

108. $x^2+y-10x+22=0$

109. $x^2+12y^2-12x+24=0$

110. $x^2+y^2+10y+22=0$

111. $4y^2-20x^2+16y+20x-9=0$

112. $16x-16y^2+24y-25=0$

113. $9x^2-9y^2-6x-18y-17=0$

114. $4x^2+4y^2+4x-8y+1=0$

Given the graph, write the equation in general form.

115. 
116. 
117. 
118. 
119. 
120. 

Solving Nonlinear Systems

Solve.

121. $$\{\begin{array}{c}{x}^2+y^2=8\\ x-y=4\end{array}$$
122. $$\{\begin{array}{c}{x}^2+y^2=1\\ x+2y=1\end{array}$$
123. $$\{\begin{array}{c}{x}^2+3y^2=4\\ 2x-y=1\end{array}$$
124. $$\{\begin{array}{c}2x^2+y^2=5\\ x+y=3\end{array}$$
125. $$\{\begin{array}{c}3x^2-2y^2=1\\ x-y=2\end{array}$$
126. $$\{\begin{array}{c}{x}^2-3y^2=10\\ x-2y=1\end{array}$$
127. $$\{\begin{array}{c}2x^2+y^2=11\\ 4x+y^2=5\end{array}$$
128. $$\{\begin{array}{c}{x}^2+4y^2=1\\ 2x^2+4y=5\end{array}$$
129. $$\{\begin{array}{c}5x^2-y^2=10\\ x^2+y=2\end{array}$$
130. $$\{\begin{array}{c}2x^2+y^2=1\\ 2x-4y^2=-3\end{array}$$
131. $$\{\begin{array}{c}{x}^2+4y^2=10\\ xy=2\end{array}$$
132. $$\{\begin{array}{c}y+x^2=0\\ xy-8=0\end{array}$$
133. $$\{\begin{array}{c}\frac{1}{x}+\frac{1}{y}=10\\ \frac{1}{x}-\frac{1}{y}=6\end{array}$$
134. $$\{\begin{array}{c}\frac{1}{x}+\frac{1}{y}=1\\ y-x=2\end{array}$$
135. $$\{\begin{array}{l}x-2y^2=3\\ y=\sqrt{x-4}\end{array}$$
136. $$\{\begin{array}{c}{\left(x-1\right)}^2+y^2=1\\ y-\sqrt{x}=0\end{array}$$

Answers

1. Distance: 5 units; midpoint: $\left(-2,\frac{1}{2}\right)$

3. Distance: $4\sqrt{10}$ units; midpoint: $\left(-4,-2\right)$

5. Distance: $\sqrt{7}$ units; midpoint: $\left(\frac{\sqrt{5}}{2},-\frac{7\sqrt{2}}{2}\right)$

7. $18\mathit{\pi }$ square units

9. $\frac{5\mathit{\pi }}{4}$ square units

11. $y={\left(x-5\right)}^2+8$; vertex: $\left(5,8\right)$

13. $y={\left(x-\frac{3}{2}\right)}^2-\frac{13}{4}$; vertex: $\left(\frac{3}{2},-\frac{13}{4}\right)$

15. $x={\left(y+5\right)}^2-15$; vertex: $\left(-15,-5\right)$

17. $x=-{\left(y-4\right)}^2+13$; vertex: $\left(13,4\right)$

19. $y={\left(x-10\right)}^2-25$;

    

21. $y=-2{\left(x+3\right)}^2-6$;

    

23. $x={\left(y-5\right)}^2-9$;

    

25. $x=-4{\left(y-\frac{3}{2}\right)}^2+9$;

    

27. $x=-4{\left(y-\frac{1}{2}\right)}^2+3$;

    

29. Center: $\left(6,0\right)$; radius: $r=3$

31. Center: $\left(0,0\right)$; radius: $r=\sqrt{5}$

33. ${\left(x+7\right)}^2+{\left(y-2\right)}^2=100$

35. $x^2+{\left(y+5\right)}^2=28$

37. ${\left(x+3\right)}^2+{\left(y-9\right)}^2=2$

39. x-intercepts: none; y-intercepts: $\left(0,-5\pm \sqrt{7}\right)$

41. x-intercepts: $\left(\pm 4,0\right)$; y-intercepts: $\left(0,2\pm 2\sqrt{5}\right)$

43. x-intercepts: none; y-intercepts: $\left(0,3\right)$, $\left(0,9\right)$

45. 
47. 
49. 

51. ${\left(x-3\right)}^2+{\left(y+2\right)}^2=16$;

    

53. ${\left(x-\frac{1}{2}\right)}^2+{\left(y-\frac{3}{2}\right)}^2=4$;

    

55. ${\left(x-\frac{5}{2}\right)}^2+{\left(y+\frac{1}{2}\right)}^2=7$;

    

57. Center: $\left(-12,10\right)$; orientation: horizontal; major radius: 4 units; minor radius: 2 units

59. Center: $\left(0,5\right)$; orientation: vertical; major radius: $2\sqrt{3}$ units; minor radius: 1 unit

61. $\frac{x^2}{9}+\frac{{\left(y+4\right)}^2}{16}=1$

63. $\frac{x^2}{25}+\frac{y^2}{2}=1$

65. x-intercepts: $\left(-4,0\right)$, $\left(0,0\right)$; y-intercepts: $\left(0,0\right)$

67. x-intercepts: $\left(\pm 2,0\right)$; y-intercepts: $\left(0,\pm \sqrt{10}\right)$

69. 
71. 
73. 

75. $\frac{{\left(x-1\right)}^2}{9}+\frac{{\left(y+5\right)}^2}{4}=1$;

    

77. $\frac{{\left(x+3\right)}^2}{9}+{\left(y+6\right)}^2=1$;

    

79. $\frac{{\left(x-1\right)}^2}{6}+\frac{{\left(y-\frac{3}{2}\right)}^2}{4}=1$;

    

81. Center: $\left(10,-5\right)$; opens left and right; vertices: $\left(8,-5\right)$, $\left(12,-5\right)$

83. Center: $\left(15,20\right)$; opens upward and downward; vertices: $\left(15,20-\sqrt{3}\right)$, $\left(15,20+\sqrt{3}\right)$

85. $\frac{{\left(y-10\right)}^2}{5}-\frac{{\left(x+25\right)}^2}{9}=1$

87. ${\left(x+4\right)}^2-\frac{y^2}{36}=1$

89. x-intercepts: $\left(1\pm 2\sqrt{2},0\right)$; y-intercepts: none

91. x-intercepts: $\left(0,0\right)$; y-intercepts: $\left(0,0\right)$, $\left(0,4\right)$

93. 
95. 
97. 

99. $\frac{{\left(x-1\right)}^2}{9}-\frac{{\left(y+5\right)}^2}{4}=1$;

    

101. $\frac{{\left(y-2\right)}^2}{2}-\frac{x^2}{25}=1$;

     

103. $\frac{{\left(y+\frac{3}{2}\right)}^2}{6}-\frac{{\left(x+\frac{1}{2}\right)}^2}{4}=1$;

     

105. Circle; ${\left(x-1\right)}^2+{\left(y-4\right)}^2=1$

107. Hyperbola; $\frac{{\left(x-3\right)}^2}{2}-\frac{{\left(y+2\right)}^2}{2}=1$

109. Ellipse; $\frac{{\left(x-6\right)}^2}{12}+y^2=1$

111. Hyperbola; $\frac{{\left(y+2\right)}^2}{5}-{\left(x-\frac{1}{2}\right)}^2=1$

113. Hyperbola; ${\left(x-\frac{1}{3}\right)}^2-{\left(y+1\right)}^2=1$

115. $x^2+y^2+18x-6y+9=0$

117. $9x^2-y^2+72x-12y+72=0$

119. $9x^2+64y^2+54x-495=0$

121. $\left(2,-2\right)$

123. $\left(-\frac{1}{13},-\frac{15}{13}\right)$, $\left(1,1\right)$

125. $\left(-9,-11\right)$, $\left(1,-1\right)$

127. $\left(-1,-3\right)$, $\left(-1,3\right)$

129. $\left(-\sqrt{2},0\right)$, $\left(\sqrt{2},0\right)$, $\left(-\sqrt{7},-5\right)$, $\left(\sqrt{7},-5\right)$

131. $\left(\sqrt{2},\sqrt{2}\right)$, $\left(-\sqrt{2},-\sqrt{2}\right)$, $\left(2\sqrt{2},\frac{\sqrt{2}}{2}\right)$, $\left(-2\sqrt{2},-\frac{\sqrt{2}}{2}\right)$

133. $\left(\frac{1}{8},\frac{1}{2}\right)$

135. $\left(5,1\right)$

Sample Exam

1. Given two points $\left(-4,-6\right)$ and $\left(2,-8\right)$:

   1. Calculate the distance between them.
   2. Find the midpoint between them.

2. Determine the area of a circle whose diameter is defined by the points $\left(4,-3\right)$ and $\left(-1,2\right).$

Rewrite in standard form and graph. Find the vertex and all intercepts if any.

3. $y=-x^2+6x-5$

4. $x=2y^2+4y-6$

5. $x=-3y^2+3y+1$

6. Find the equation of a circle in standard form with center $\left(-6,3\right)$ and radius $2\sqrt{5}$ units.

Sketch the graph of the conic section given its equation in standard form.

7. ${\left(x-4\right)}^2+{\left(y+1\right)}^2=45$

8. $\frac{{\left(x+3\right)}^2}{4}+\frac{y^2}{9}=1$

9. $\frac{y^2}{3}-\frac{x^2}{9}=1$

10. $\frac{x^2}{16}-{\left(y-2\right)}^2=1$

Rewrite in standard form and graph.

11. $9x^2+4y^2-144x+16y+556=0$

12. $x-y^2+6y+7=0$

13. $x^2+y^2+20x-20y+100=0$

14. $4y^2-x^2+40y-30x-225=0$

Find the x- and y-intercepts.

15. $x=-2{\left(y-4\right)}^2+9$

16. $\frac{{\left(y-1\right)}^2}{12}-{\left(x+1\right)}^2=1$

Solve.

17. $$\{\begin{array}{l}x+y=2\\ y=-x^2+4\end{array}$$
18. $$\{\begin{array}{c}y-x^2=-3\\ x^2+y^2=9\end{array}$$
19. $$\{\begin{array}{c}2x-y=1\\ {\left(x+1\right)}^2+2y^2=1\end{array}$$
20. $$\{\begin{array}{c}{x}^2+y^2=6\\ xy=3\end{array}$$

21. Find the equation of an ellipse in standard form with vertices $\left(-3,-5\right)$ and $\left(5,-5\right)$ and a minor radius 2 units in length.

22. Find the equation of a hyperbola in standard form opening left and right with vertices $\left(\pm \sqrt{5},0\right)$ and a conjugate axis that measures 10 units.

23. Given the graph of the ellipse, determine its equation in general form.

    

24. A rectangular deck has an area of 80 square feet and a perimeter that measures 36 feet. Find the dimensions of the deck.

25. The diagonal of a rectangle measures $2\sqrt{13}$ centimeters and the perimeter measures 20 centimeters. Find the dimensions of the rectangle.

Answers

1. 1. $2\sqrt{10}$ units;
   2. $\left(-1,-7\right)$

3. $y=-{\left(x-3\right)}^2+4$;

   

5. $x=-3{\left(y-\frac{1}{2}\right)}^2+\frac{7}{4}$;

   
7. 
9. 

11. $\frac{{\left(x-8\right)}^2}{4}+\frac{{\left(y+2\right)}^2}{9}=1$;

    

13. ${\left(x+10\right)}^2+{\left(y-10\right)}^2=100$;

    

15. x-intercept: $\left(-23,0\right)$; y-intercepts: $\left(0,\frac{8\pm 3\sqrt{2}}{2}\right)$

17. $\left(-1,3\right)$, $\left(2,0\right)$

19. Ø

21. $\frac{{\left(x-1\right)}^2}{16}+\frac{{\left(y+5\right)}^2}{4}=1$

23. $4x^2+25y^2-24x-100y+36=0$

25. 6 centimeters by 4 centimeters