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

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?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali