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

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

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

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

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

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: $\text{\hspace{0.17em}}x=3.$

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
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
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert