# 8.5 Conic sections in polar coordinates  (Page 6/8)

 Page 6 / 8

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

$\frac{{\left(y-1\right)}^{2}}{49}-\frac{{\left(x+1\right)}^{2}}{4}=1$

${x}^{2}-4{y}^{2}+6x+32y-91=0$

$2{y}^{2}-{x}^{2}-12y-6=0$

For the following exercises, find the equation of the hyperbola.

Center at $\text{\hspace{0.17em}}\left(0,0\right),$ vertex at $\text{\hspace{0.17em}}\left(0,4\right),$ focus at $\text{\hspace{0.17em}}\left(0,-6\right)$

Foci at $\text{\hspace{0.17em}}\left(3,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(7,7\right),$ vertex at $\text{\hspace{0.17em}}\left(6,7\right)$

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

## The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

${y}^{2}=12x$

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

${\left(x+2\right)}^{2}=\frac{1}{2}\left(y-1\right);\text{\hspace{0.17em}}$ vertex: $\text{\hspace{0.17em}}\left(-2,1\right);\text{\hspace{0.17em}}$ focus: $\text{\hspace{0.17em}}\left(-2,\frac{9}{8}\right);\text{\hspace{0.17em}}$ directrix: $\text{\hspace{0.17em}}y=\frac{7}{8}$

${y}^{2}-6y-6x-3=0$

${x}^{2}+10x-y+23=0$

${\left(x+5\right)}^{2}=\left(y+2\right);\text{\hspace{0.17em}}$ vertex: $\text{\hspace{0.17em}}\left(-5,-2\right);\text{\hspace{0.17em}}$ focus: $\text{\hspace{0.17em}}\left(-5,-\frac{7}{4}\right);\text{\hspace{0.17em}}$ directrix: $\text{\hspace{0.17em}}y=-\frac{9}{4}$

For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

${x}^{2}+4y=0$

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

${x}^{2}-8x-10y+46=0$

$2{y}^{2}+12y+6x+15=0$

For the following exercises, write the equation of the parabola using the given information.

Focus at $\text{\hspace{0.17em}}\left(-4,0\right);\text{\hspace{0.17em}}$ directrix is $\text{\hspace{0.17em}}x=4$

Focus at $\text{\hspace{0.17em}}\left(2,\frac{9}{8}\right);\text{\hspace{0.17em}}$ directrix is $\text{\hspace{0.17em}}y=\frac{7}{8}$

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

A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

## Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

$16{x}^{2}+24xy+9{y}^{2}+24x-60y-60=0$

${B}^{2}-4AC=0,$ parabola

$4{x}^{2}+14xy+5{y}^{2}+18x-6y+30=0$

$4{x}^{2}+xy+2{y}^{2}+8x-26y+9=0$

${B}^{2}-4AC=-31<0,$ ellipse

For the following exercises, determine the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that will eliminate the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term, and write the corresponding equation without the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term.

${x}^{2}+4xy-2{y}^{2}-6=0$

${x}^{2}-xy+{y}^{2}-6=0$

$\theta ={45}^{\circ },{{x}^{\prime }}^{2}+3{{y}^{\prime }}^{2}-12=0$

For the following exercises, graph the equation relative to the $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ system in which the equation has no $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ term.

$9{x}^{2}-24xy+16{y}^{2}-80x-60y+100=0$

${x}^{2}-xy+{y}^{2}-2=0$

$\theta ={45}^{\circ }$

$6{x}^{2}+24xy-{y}^{2}-12x+26y+11=0$

## Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

Hyperbola with $\text{\hspace{0.17em}}e=5\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ units to the left of the pole.

Ellipse with $\text{\hspace{0.17em}}e=\frac{3}{4}\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{1}{3}\text{\hspace{0.17em}}$ unit above the pole.

For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.

For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

Directrix is $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and eccentricity $\text{\hspace{0.17em}}e=1$

Directrix is $\text{\hspace{0.17em}}y=-2\text{\hspace{0.17em}}$ and eccentricity $\text{\hspace{0.17em}}e=4$

## Practice test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

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

$\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1;\text{\hspace{0.17em}}$ center: $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(3,0\right),\left(–3,0\right),\left(0,2\right),\left(0,-2\right);\text{\hspace{0.17em}}$ foci: $\left(\sqrt{5},0\right),\left(-\sqrt{5},0\right)$

$9{y}^{2}+16{x}^{2}-36y+32x-92=0$

For the following exercises, sketch the graph, identifying the center, vertices, and foci.

$\frac{{\left(x-3\right)}^{2}}{64}+\frac{{\left(y-2\right)}^{2}}{36}=1$

center: $\text{\hspace{0.17em}}\left(3,2\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(11,2\right),\left(-5,2\right),\left(3,8\right),\left(3,-4\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(3+2\sqrt{7},2\right),\left(3-2\sqrt{7},2\right)$

$2{x}^{2}+{y}^{2}+8x-6y-7=0$

Write the standard form equation of an ellipse with a center at $\text{\hspace{0.17em}}\left(1,2\right),$ vertex at $\text{\hspace{0.17em}}\left(7,2\right),$ and focus at $\text{\hspace{0.17em}}\left(4,2\right).$

$\frac{{\left(x-1\right)}^{2}}{36}+\frac{{\left(y-2\right)}^{2}}{27}=1$

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

$\frac{{x}^{2}}{49}-\frac{{y}^{2}}{81}=1$

$\frac{{x}^{2}}{{7}^{2}}-\frac{{y}^{2}}{{9}^{2}}=1;\text{\hspace{0.17em}}$ center: $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ vertices $\text{\hspace{0.17em}}\left(7,0\right),\left(-7,0\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(\sqrt{130},0\right),\left(-\sqrt{130},0\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=±\frac{9}{7}x$

$16{y}^{2}-9{x}^{2}+128y+112=0$

For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

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

center: $\text{\hspace{0.17em}}\left(3,-3\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(8,-3\right),\left(-2,-3\right);$ foci: $\text{\hspace{0.17em}}\left(3+\sqrt{26},-3\right),\left(3-\sqrt{26},-3\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=±\frac{1}{5}\left(x-3\right)-3$

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

Write the standard form equation of a hyperbola with foci at $\text{\hspace{0.17em}}\left(1,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1,6\right),$ and a vertex at $\text{\hspace{0.17em}}\left(1,2\right).$

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

For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

${y}^{2}+10x=0$

$3{x}^{2}-12x-y+11=0$

${\left(x-2\right)}^{2}=\frac{1}{3}\left(y+1\right);\text{\hspace{0.17em}}$ vertex: $\text{\hspace{0.17em}}\left(2,-1\right);\text{\hspace{0.17em}}$ focus: $\text{\hspace{0.17em}}\left(2,-\frac{11}{12}\right);\text{\hspace{0.17em}}$ directrix: $\text{\hspace{0.17em}}y=-\frac{13}{12}$

For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

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

${y}^{2}+8x-8y+40=0$

Write the equation of a parabola with a focus at $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}y=-1.$

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Approximately $\text{\hspace{0.17em}}8.49\text{\hspace{0.17em}}$ feet

For the following exercises, determine which conic section is represented by the given equation, and then determine the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that will eliminate the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term.

$3{x}^{2}-2xy+3{y}^{2}=4$

${x}^{2}+4xy+4{y}^{2}+6x-8y=0$

parabola; $\text{\hspace{0.17em}}\theta \approx {63.4}^{\circ }$

For the following exercises, rewrite in the $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ system without the $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ term, and graph the rotated graph.

$11{x}^{2}+10\sqrt{3}xy+{y}^{2}=4$

$16{x}^{2}+24xy+9{y}^{2}-125x=0$

${{x}^{\prime }}^{2}-4{x}^{\prime }+3{y}^{\prime }=0$

For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

Hyperbola with $\text{\hspace{0.17em}}e=\frac{3}{2},\text{\hspace{0.17em}}$ and directrix $\text{\hspace{0.17em}}\frac{5}{6}\text{\hspace{0.17em}}$ units to the right of the pole.

For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.

Find a polar equation of the conic with focus at the origin, eccentricity of $\text{\hspace{0.17em}}e=2,$ and directrix: $\text{\hspace{0.17em}}x=3.$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
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
hi vedant can u help me with some assignments
Solomon
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
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
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
how
Sheref
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
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salma
Commplementary angles
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Sherica
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Sherica
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Tamia
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Uday
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Ali
greetings from Iran
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Nharnhar