15.2.2: Homework — Read — Invisible Network

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Reading Questions

Define a hyperbola in terms of its foci. How does this definition differ from the definition of an ellipse?
What is the transverse axis of a hyperbola? How can you identify which axis is the transverse axis from the standard form equation?
What is the role of the "central rectangle" in graphing a hyperbola?
What is the equation relating the distances from the center to the vertices ($v_x, v_y$) and the distance from the center to the foci ($c$)? How does this differ from the corresponding equation for an ellipse?
What is the standard form of the equation for a hyperbola with a horizontal transverse axis, centered at $(h,k)$?
What is the formula for the slopes of the asymptotes of a hyperbola centered at $(h,k)$?
If the asymptotes of a hyperbola intersect at the origin, what can you conclude about the hyperbola?
Do the foci of a hyperbola lie on the transverse axis or the conjugate axis?
If the transverse axis of a hyperbola is vertical, what does this tell you about its graph and its standard form equation?
Can the graph of a hyperbola ever cross its asymptotes? Can it cross its transverse axis? Can it cross its conjugate axis?
Define a hyperbola in terms of its foci.
What can we conclude about a hyperbola if its asymptotes intersect at the origin?
What must be true of the foci of a hyperbola?
If the transverse axis of a hyperbola is vertical, what do we know about the graph?
Where must the center of hyperbola be relative to its foci?

Homework

Algebraic

For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.

$ 3y^2+2x=6$
$ \frac{x^2}{36}−\frac{y^2}{9}=1$
$ 5y^2+4x^2=6x$
$25x^2−16y^2=400$
$−9x^2+18x+y^2+4y−14=0$

For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$ \frac{x^2}{25} − \frac{y^2}{36}=1$
$ \frac{x^2}{100}−\frac{y^2}{9}=1 $
$ \frac{y^2}{4}−\frac{x^2}{81}=1 $
$ 9y^2−4x^2=1 $
$ \frac{(x−1)^2}{9}−\frac{(y−2)^2}{16}=1 $
$ \frac{(y−6)^2}{36}−\frac{(x+1)^2}{16}=1 $
$ \frac{(x−2)^2}{49}−\frac{(y+7)^2}{49}=1 $
$ 4x^2−8x−9y^2−72y+112=0 $
$ −9x^2−54x+9y^2−54y+81=0 $
$ 4x^2−24x−36y^2−360y+864=0 $
$ −4x^2+24x+16y^2−128y+156=0 $
$ −4x^2+40x+25y^2−100y+100=0 $
$ x^2+2x−100y^2−1000y+2401=0 $
$ −9x^2+72x+16y^2+16y+4=0 $
$ 4x^2+24x−25y^2+200y−464=0 $

For the following exercises, find the equations of the asymptotes for each hyperbola.

$ \frac{y^2}{3^2}−\frac{x^2}{3^2}=1 $
$ \frac{(x−3)^2}{5^2}−\frac{(y+4)^2}{2^2} = 1 $
$ \frac{(y−3)^2}{3^2}−\frac{(x+5)^2}{6^2}=1 $
$ 9x^2−18x−16y^2+32y−151=0 $
$ 16y^2+96y−4x^2+16x+112=0 $

Graphical

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.

$ \frac{x^2}{49}−\frac{y^2}{16}=1 $
$ \frac{x^2}{64}−\frac{y^2}{4}=1 $
$ \frac{y^2}{9}−\frac{x^2}{25}=1 $
$ 81x^2−9y^2=1 $
$ \frac{(y+5)^2}{9}−\frac{(x−4)^2}{25}=1 $
$ \frac{(x−2)^2}{8}−\frac{(y+3)^2}{27}=1 $
$ \frac{(y−3)^2}{9}−\frac{(x−3)^2}{9}=1 $
$ −4x^2−8x+16y^2−32y−52=0 $
$ x^2−8x−25y^2−100y−109=0 $
$ −x^2+8x+4y^2−40y+88=0 $
$ 64x^2+128x−9y^2−72y−656=0 $
$ 16x^2+64x−4y^2−8y−4=0 $
$ −100x^2+1000x+y^2−10y−2575=0 $
$ 4x^2+16x−4y^2+16y+16=0 $

For the following exercises, given information about the graph of the hyperbola, find its equation.

Vertices at $ (3,0) $ and $ (−3,0) $ and one focus at $ (5,0) $.
Vertices at $ (0,6) $ and $ (0,−6) $ and one focus at $ (0,−8) $.
Vertices at $ (1,1) $ and $ (11,1) $ and one focus at $ (12,1) $.
Center: $ (0,0) $; vertex: $ (0,−13) $; one focus: $ (0, \sqrt{313}) $.
Center: $ (4,2) $; vertex: $ (9,2) $; one focus: $ (4+\sqrt{26},2) $.
Center: $ (3,5) $; vertex: $ (3,11) $; one focus: $ (3,5+2\sqrt{10}) $.

For the following exercises, given the graph of the hyperbola, find its equation.

$A vertical hyperbola centered at (0, 0) with vertices at (0, negative 4) and (0, 4). The slant asymptotes are shown but not labeled.$
$A horizontal hyperbola centered at (1, 1) with vertices at (1 minus square root of 2, 1) and (1 + square root of 2, 1) and foci at (1 minus square root of 5, 1) and (1 + square root of 5, 1)$
$A vertical hyperbola centered at (negative 1, 0) with vertices at (negative 1, negative 3) and (negative 1, 3) and foci at (negative 1, negative 3 square root of 2) and (negative 1, 3 square root of 2).$
$A vertical hyperbola centered at (3, 1) with vertices at (3, 1 minus square root of 2) and (3, 1 + square root of 2) and foci at (3, 1 minus square root of 7) and (3, 1 + square root of 7).$
$A horizontal hyperbola centered at (negative 3, negative 3) with vertices at (negative 8, negative 3) and (2, negative 3) and foci at (negative 3 minus 5 square root of 2, negative 3) and (negative 3 + 5 square root of 2, negative 3).$

Extensions

For the following exercises, express the equation for the hyperbola as two functions, with $ y $ as a function of $ x $. Express as simply as possible. Use graphing technology to sketch the graph of the two functions on the same axes.

$ \frac{x^2}{4}−\frac{y^2}{9}=1 $
$ \frac{y^2}{9}−x^2=1 $
$ \frac{(x−2)^2}{16}−\frac{(y+3)^2}{25}=1 $
$ −4x^2−16x+y^2−2y−19=0 $
$ 4x^2−24x−y^2−4y+16=0 $

Real-World Applications

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.

The hedge will follow the asymptotes $ y=x $ and $ y=−x $, and its closest distance to the center fountain is 5 yards.
The hedge will follow the asymptotes $ y=2x $ and $ y=−2x $, and its closest distance to the center fountain is 6 yards.
The hedge will follow the asymptotes $ y=\frac{1}{2}x $ and $ y=−\frac{1}{2}x $, and its closest distance to the center fountain is 10 yards.
The hedge will follow the asymptotes $ y = \frac{2}{3}x $ and $ y= −\frac{2}{3}x$, and its closest distance to the center fountain is 12 yards.
The hedge will follow the asymptotes $ y = \frac{3}{4}x $ and $ y = -\frac{3}{4}x $, and its closest distance to the center fountain is 20 yards.

For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the $ x $-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.

The object enters along a path approximated by the line $ y=x−2 $ and passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $ y=−x+2 $.
The object enters along a path approximated by the line $ y=2x−2 $ and passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $ y=−2x+2 $.
The object enters along a path approximated by the line $ y=0.5x+2 $ and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $ y=−0.5x−2 $.
The object enters along a path approximated by the line $ y=\frac{1}{3}x−1 $ and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $ y=−\frac{1}{3}x+1 $.
The object enters along a path approximated by the line $ y=3x−9 $ and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line $ y=−3x+9 $.

15.2.2: Homework is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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