# 8.2 The hyperbola  (Page 8/13)

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A design for a cooling tower project is shown in [link] . Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

The sides of the tower can be modeled by the hyperbolic equation.

Access these online resources for additional instruction and practice with hyperbolas.

## Key equations

 Hyperbola, center at origin, transverse axis on x -axis $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ Hyperbola, center at origin, transverse axis on y -axis $\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$ Hyperbola, center at $\text{\hspace{0.17em}}\left(h,k\right),$ transverse axis parallel to x -axis $\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$ Hyperbola, center at $\text{\hspace{0.17em}}\left(h,k\right),$ transverse axis parallel to y -axis $\frac{{\left(y-k\right)}^{2}}{{a}^{2}}-\frac{{\left(x-h\right)}^{2}}{{b}^{2}}=1$

## Key concepts

• A hyperbola is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ in a plane such that the difference of the distances between $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ and the foci is a positive constant.
• The standard form of a hyperbola can be used to locate its vertices and foci. See [link] .
• When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See [link] and [link] .
• When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See [link] and [link] .
• Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See [link] .

## Verbal

Define a hyperbola in terms of its foci.

A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

What can we conclude about a hyperbola if its asymptotes intersect at the origin?

What must be true of the foci of a hyperbola?

The foci must lie on the transverse axis and be in the interior of the 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?

The center must be the midpoint of the line segment joining the foci.

## Algebraic

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

$3{y}^{2}+2x=6$

$\frac{{x}^{2}}{36}-\frac{{y}^{2}}{9}=1$

yes $\text{\hspace{0.17em}}\frac{{x}^{2}}{{6}^{2}}-\frac{{y}^{2}}{{3}^{2}}=1$

$5{y}^{2}+4{x}^{2}=6x$

$25{x}^{2}-16{y}^{2}=400$

yes $\text{\hspace{0.17em}}\frac{{x}^{2}}{{4}^{2}}-\frac{{y}^{2}}{{5}^{2}}=1$

$-9{x}^{2}+18x+{y}^{2}+4y-14=0$

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.

$\frac{{x}^{2}}{25}-\frac{{y}^{2}}{36}=1$

$\frac{{x}^{2}}{{5}^{2}}-\frac{{y}^{2}}{{6}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(5,0\right),\left(-5,0\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(\sqrt{61},0\right),\left(-\sqrt{61},0\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=\frac{6}{5}x,y=-\frac{6}{5}x\text{\hspace{0.17em}}$

$\frac{{x}^{2}}{100}-\frac{{y}^{2}}{9}=1$

$\frac{{y}^{2}}{4}-\frac{{x}^{2}}{81}=1$

$\frac{{y}^{2}}{{2}^{2}}-\frac{{x}^{2}}{{9}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(0,2\right),\left(0,-2\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(0,\sqrt{85}\right),\left(0,-\sqrt{85}\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=\frac{2}{9}x,y=-\frac{2}{9}x$

$9{y}^{2}-4{x}^{2}=1$

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

$\frac{{\left(x-1\right)}^{2}}{{3}^{2}}-\frac{{\left(y-2\right)}^{2}}{{4}^{2}}=1;\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(4,2\right),\left(-2,2\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(6,2\right),\left(-4,2\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=\frac{4}{3}\left(x-1\right)+2,y=-\frac{4}{3}\left(x-1\right)+2$

the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al