# 12.5 Conic sections in polar coordinates  (Page 5/8)

 Page 5 / 8

For the following exercises, convert the polar equation of a conic section to a rectangular equation.

$25{x}^{2}+16{y}^{2}-12y-4=0$

$21{x}^{2}-4{y}^{2}-30x+9=0$

$64{y}^{2}=48x+9$

$96{y}^{2}-25{x}^{2}+110y+25=0$

$3{x}^{2}+4{y}^{2}-2x-1=0$

$5{x}^{2}+9{y}^{2}-24x-36=0$

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

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

Directrix: $x=4;\text{\hspace{0.17em}}e=\frac{1}{5}$

$r=\frac{4}{5+\mathrm{cos}\theta }$

Directrix: $x=-4;\text{\hspace{0.17em}}e=5$

Directrix: $y=2;\text{\hspace{0.17em}}e=2$

$r=\frac{4}{1+2\mathrm{sin}\theta }$

Directrix: $y=-2;\text{\hspace{0.17em}}e=\frac{1}{2}$

Directrix: $x=1;\text{\hspace{0.17em}}e=1$

$r=\frac{1}{1+\mathrm{cos}\theta }$

Directrix: $x=-1;\text{\hspace{0.17em}}e=1$

Directrix: $x=-\frac{1}{4};\text{\hspace{0.17em}}e=\frac{7}{2}$

$r=\frac{7}{8-28\mathrm{cos}\theta }$

Directrix: $y=\frac{2}{5};\text{\hspace{0.17em}}e=\frac{7}{2}$

Directrix: $y=4;\text{\hspace{0.17em}}e=\frac{3}{2}$

$r=\frac{12}{2+3\mathrm{sin}\theta }$

Directrix: $x=-2;\text{\hspace{0.17em}}e=\frac{8}{3}$

Directrix: $x=-5;\text{\hspace{0.17em}}e=\frac{3}{4}$

$r=\frac{15}{4-3\mathrm{cos}\theta }$

Directrix: $y=2;\text{\hspace{0.17em}}e=2.5$

Directrix: $x=-3;\text{\hspace{0.17em}}e=\frac{1}{3}$

$r=\frac{3}{3-3\mathrm{cos}\theta }$

## Extensions

Recall from Rotation of Axes that equations of conics with an $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term have rotated graphs. For the following exercises, express each equation in polar form with $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}\theta .$

$xy=2$

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

$r=±\frac{2}{\sqrt{1+\mathrm{sin}\theta \mathrm{cos}\theta }}$

$2{x}^{2}+4xy+2{y}^{2}=9$

$16{x}^{2}+24xy+9{y}^{2}=4$

$r=±\frac{2}{4\mathrm{cos}\theta +3\mathrm{sin}\theta }$

$2xy+y=1$

## The Ellipse

For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

$\frac{{x}^{2}}{25}+\frac{{y}^{2}}{64}=1$

$\frac{{x}^{2}}{{5}^{2}}+\frac{{y}^{2}}{{8}^{2}}=1;\text{\hspace{0.17em}}$ center: $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(5,0\right),\left(-5,0\right),\left(0,8\right),\left(0,-8\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(0,\sqrt{39}\right),\left(0,-\sqrt{39}\right)$

$\frac{{\left(x-2\right)}^{2}}{100}+\frac{{\left(y+3\right)}^{2}}{36}=1$

$9{x}^{2}+{y}^{2}+54x-4y+76=0$

$\frac{{\left(x+3\right)}^{2}}{{1}^{2}}+\frac{{\left(y-2\right)}^{2}}{{3}^{2}}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(-3,2\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(-2,2\right),\left(-4,2\right),\left(-3,5\right),\left(-3,-1\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(-3,2+2\sqrt{2}\right),\left(-3,2-2\sqrt{2}\right)$

$9{x}^{2}+36{y}^{2}-36x+72y+36=0$

For the following exercises, graph the ellipse, noting center, vertices, and foci.

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

center: $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(6,0\right),\left(-6,0\right),\left(0,3\right),\left(0,-3\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(3\sqrt{3},0\right),\left(-3\sqrt{3},0\right)$

$\frac{{\left(x-4\right)}^{2}}{25}+\frac{{\left(y+3\right)}^{2}}{49}=1$

$4{x}^{2}+{y}^{2}+16x+4y-44=0$

center: $\text{\hspace{0.17em}}\left(-2,-2\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(2,-2\right),\left(-6,-2\right),\left(-2,6\right),\left(-2,-10\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(-2,-2+4\sqrt{3},\right),\left(-2,-2-4\sqrt{3}\right)$

$\text{\hspace{0.17em}}2{x}^{2}+3{y}^{2}-20x+12y+38=0$

For the following exercises, use the given information to find the equation for the ellipse.

Center at $\text{\hspace{0.17em}}\left(0,0\right),$ focus at $\text{\hspace{0.17em}}\left(3,0\right),$ vertex at $\text{\hspace{0.17em}}\left(-5,0\right)$

$\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$

Center at $\text{\hspace{0.17em}}\left(2,-2\right),$ vertex at $\text{\hspace{0.17em}}\left(7,-2\right),$ focus at $\text{\hspace{0.17em}}\left(4,-2\right)$

A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

Approximately 35.71 feet

## The Hyperbola

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

$\frac{{x}^{2}}{81}-\frac{{y}^{2}}{9}=1$

$\frac{{\left(y+1\right)}^{2}}{16}-\frac{{\left(x-4\right)}^{2}}{36}=1$

$\frac{{\left(y+1\right)}^{2}}{{4}^{2}}-\frac{{\left(x-4\right)}^{2}}{{6}^{2}}=1;\text{\hspace{0.17em}}$ center: $\text{\hspace{0.17em}}\left(4,-1\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(4,3\right),\left(4,-5\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(4,-1+2\sqrt{13}\right),\left(4,-1-2\sqrt{13}\right)$

$9{y}^{2}-4{x}^{2}+54y-16x+29=0$

$3{x}^{2}-{y}^{2}-12x-6y-9=0$

$\frac{{\left(x-2\right)}^{2}}{{2}^{2}}-\frac{{\left(y+3\right)}^{2}}{{\left(2\sqrt{3}\right)}^{2}}=1;\text{\hspace{0.17em}}$ center: $\text{\hspace{0.17em}}\left(2,-3\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(4,-3\right),\left(0,-3\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(6,-3\right),\left(-2,-3\right)$

For the following exercises, graph the hyperbola, labeling vertices and foci.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions