# 10.2 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: $\text{\hspace{0.17em}}y=\frac{4}{3}\left(x-1\right)+2,y=-\frac{4}{3}\left(x-1\right)+2$

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
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how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
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how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim