A hyperbola is the set of all points in a plane such that the absolute value of the difference between two fixed points is constant. The two fixed points are known as the foci. In this section, we learned how to sketch the graph of a hyperbola and find the vertices, foci, asymptotes, and endpoints for the fundamental rectangle.

Test Objectives
• Demonstrate the ability to write the equation of a hyperbola in standard form
• Demonstrate the ability to find the foci of a hyperbola
• Demonstrate the ability to find the vertices of a hyperbola
• Demonstrate the ability to sketch the graph of a hyperbola
Conic Sections: The Hyperbola Practice Test:

#1:

Instructions: Sketch the graph of each, and state the foci and vertices.

$$a)\hspace{.2em}\frac{(y + 1)^2}{16}- \frac{(x - 2)^2}{9}=1$$

$$b)\hspace{.2em}\frac{(x - 1)^2}{9}- \frac{(y + 2)^2}{4}=1$$

#2:

Instructions: Write each in standard form and then sketch the graph.

$$a)\hspace{.2em}{-}4x^2 + 9y^2 - 16x - 126y - 151=0$$

$$b)\hspace{.2em}{-}x^2 + y^2 + 20x + 8y - 120=0$$

#3:

Instructions: Write each in standard form and then sketch the graph.

$$a)\hspace{.2em}4x^2 - y^2 + 56x - 4y + 92=0$$

$$b)\hspace{.2em}{-}x^2 + 4y^2 + 2x - 64y + 239=0$$

#4:

Instructions: Write each in standard form and then sketch the graph.

$$a)\hspace{.2em}\text{Vertices:} \, (7, 17), (7, 3)$$ $$\text{Foci:} \, \left(7, 10 + \sqrt{149}\right), \left(7, 10 - \sqrt{149}\right)$$

$$b)\hspace{.2em}\text{Vertices:} \, (21, -9), (-3, -9)$$ $$\text{Foci:} \, (24, -9), (-6, -9)$$

#5:

Instructions: Write each in standard form and then sketch the graph.

$$a)\hspace{.2em}\text{Vertices:} \, (19, -4), (-5, -4)$$ $$\text{Foci:} \, (20, -4), (-6, -4)$$

$$b)\hspace{.2em}\text{Vertices:} \, (-4, 21), (-4, -9)$$ $$\text{Foci:} \, (-4, 23), (-4, -11)$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ $$\text{Vertices:}\, (2, 3), (2, -5)$$ $$\text{Foci:} \, (2, 4), (2, -6)$$ $$\text{Asymptotes:}$$ $$y = \frac{4}{3}x - \frac{11}{3}$$ $$y = -\frac{4}{3}x + \frac{5}{3}$$ Desmos Link for More Detail

$$\frac{(y + 1)^2}{16} - \frac{(x - 2)^2}{9} = 1$$

$$b)\hspace{.2em}$$ $$\text{Vertices:}\, (4, -2), (-2, -2)$$ $$\text{Foci:} \, (1 + \sqrt{13}, -2), (1 - \sqrt{13}, -2)$$ Note for the graph: $$1 + \sqrt{13} ≈ 4.606$$ $$1 - \sqrt{13} ≈ -2.606$$ $$\text{Asymptotes:}$$ $$y = \frac{2}{3}x - \frac{8}{3}$$ $$y = -\frac{2}{3}x - \frac{4}{3}$$ Desmos Link for More Detail

$$\frac{(x - 1)^2}{9} - \frac{(y + 2)^2}{4} = 1$$

#2:

Solutions:

$$a)\hspace{.2em}\frac{(y - 7)^2}{64}- \frac{(x + 2)^2}{144}=1$$ $$\text{Vertices:}\, (-2, 15), (-2, -1)$$ $$\text{Foci:} \, \left(-2, 7 + 4\sqrt{13}\right), \left(-2, 7 - 4\sqrt{13}\right)$$ Note for the graph: $$7 + 4\sqrt{13} ≈ 21.422$$ $$7 - 4\sqrt{13} ≈ -7.422$$ $$\text{Asymptotes:}$$ $$y = \frac{2}{3}x + \frac{25}{3}$$ $$y = -\frac{2}{3}x + \frac{17}{3}$$ Desmos Link for More Detail

$$\frac{(y - 7)^2}{64}- \frac{(x + 2)^2}{144}=1$$

$$b)\hspace{.2em}\frac{(y + 4)^2}{36}- \frac{(x - 10)^2}{36}=1$$ $$\text{Vertices:}\, (10, 2), (10, -10)$$ $$\text{Foci:} \, \left(10, -4 + 6\sqrt{2}\right), \left(10, -4 - 6\sqrt{2}\right)$$ Note for the graph: $$-4 + 6\sqrt{2} ≈ 4.485$$ $$-4 - 6\sqrt{2} ≈ -12.485$$ $$\text{Asymptotes:}$$ $$y = x - 14$$ $$y = -x + 6$$ Desmos Link for More Detail

$$\frac{(y + 4)^2}{36}- \frac{(x - 10)^2}{36}=1$$

#3:

Solutions:

$$a)\hspace{.2em}\frac{(x + 7)^2}{25}- \frac{(y + 2)^2}{100}=1$$ $$\text{Vertices:}\, (-2,-2), (-12,-2)$$ $$\text{Foci:} \, \left(-7 + 5\sqrt{5}, -2\right), \left(-7 - 5\sqrt{5}, -2\right)$$ Note for the graph: $$-7 + 5\sqrt{5} ≈ 4.18$$ $$-7 - 5\sqrt{5} ≈ -18.18$$ $$\text{Asymptotes:}$$ $$y = 2x + 12$$ $$y = -2x - 16$$ Desmos Link for More Detail

$$\frac{(x + 7)^2}{25}- \frac{(y + 2)^2}{100}=1$$

$$b)\hspace{.2em}\frac{(y - 8)^2}{4}- \frac{(x - 1)^2}{16}=1$$ $$\text{Vertices:}\, (1,10), (1, 6)$$ $$\text{Foci:} \, \left(1, 8 + 2\sqrt{5}\right), \left(1, 8 - 2\sqrt{5}\right)$$ Note for the graph: $$8 + 2\sqrt{5} ≈ 12.472$$ $$8 - 2\sqrt{5} ≈ 3.528$$ $$\text{Asymptotes:}$$ $$y = \frac{1}{2}x + \frac{15}{2}$$ $$y = -\frac{1}{2}x + \frac{17}{2}$$ Desmos Link for More Detail

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

#4:

Solutions:

$$a)\hspace{.2em}\frac{(y - 10)^2}{49}- \frac{(x - 7)^2}{100}=1$$ $$\text{Vertices:}\, (7, 17), (7, 3)$$ $$\text{Foci:} \, \left(7, 10 + \sqrt{149}\right), \left(7, 10 - \sqrt{149}\right)$$ Note for the graph: $$10 + \sqrt{149} ≈ 22.207$$ $$10 - \sqrt{149} ≈ -2.207$$ $$\text{Asymptotes:}$$ $$y = \frac{7}{10}x + \frac{51}{10}$$ $$y = -\frac{7}{10}x + \frac{149}{10}$$ Desmos Link for More Detail

$$\frac{(y - 10)^2}{49}- \frac{(x - 7)^2}{100}=1$$

$$b)\hspace{.2em}\frac{(x - 9)^2}{144}- \frac{(y + 9)^2}{81}=1$$ $$\text{Vertices:}\, (21, -9), (-3, -9)$$ $$\text{Foci:} \, (24, -9), (-6, -9)$$ $$\text{Asymptotes:}$$ $$y = \frac{3}{4}x - \frac{63}{4}$$ $$y = -\frac{3}{4}x - \frac{9}{4}$$ Desmos Link for More Detail

$$\frac{(x - 9)^2}{144}- \frac{(y + 9)^2}{81}=1$$

#5:

Solutions:

$$a)\hspace{.2em}\frac{(x - 7)^2}{144}- \frac{(y + 4)^2}{25}=1$$ $$\text{Vertices:}\, (19, -4), (-5, -4)$$ $$\text{Foci:} \, (20, -4), (-6, -4)$$ $$\text{Asymptotes:}$$ $$y = \frac{5}{12}x - \frac{83}{12}$$ $$y = -\frac{5}{12}x - \frac{13}{12}$$ Desmos Link for More Detail

$$\frac{(x - 7)^2}{144}- \frac{(y + 4)^2}{25}=1$$

$$b)\hspace{.2em}\frac{(y - 6)^2}{225}- \frac{(x + 4)^2}{64}=1$$ $$\text{Vertices:}\, (-4, 21), (-4, -9)$$ $$\text{Foci:} \, (-4, 23), (-4, -11)$$ $$\text{Asymptotes:}$$ $$y = \frac{15}{8}x + \frac{27}{2}$$ $$y = -\frac{15}{8}x - \frac{3}{2}$$ Desmos Link for More Detail

$$\frac{(y - 6)^2}{225}- \frac{(x + 4)^2}{64}=1$$