Section 1
The Ellipse
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{9}+\frac{y^{2}}{36}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{16}+\frac{y^{2}}{49}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{25}+\frac{y^{2}}{64}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{49}+\frac{y^{2}}{36}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{49}+\frac{y^{2}}{81}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{64}+\frac{y^{2}}{100}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{\frac{2}{4}}+\frac{y^{2}}{\frac{25}{4}}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$\frac{x^{2}}{\frac{81}{4}}+\frac{y^{2}}{\frac{25}{16}}=1$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$x^{2}=1-4 y^{2}$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$y^{2}=1-4 x^{2}$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$25 x^{2}+4 y^{2}=100$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$9 x^{2}+4 y^{2}=36$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$4 x^{2}+16 y^{2}=64$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$4 x^{2}+25 y^{2}=100$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$7 x^{2}=35-5 y^{2}$$
In Exercises $1-18,$ graph each ellipse and locate the foci.$$6 x^{2}=30-5 y^{2}$$
In Exercises $19-24,$ find the standard form of the equation of each ellipse and give the location of its foci.(GRAPH NOT COPY)
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Foci: $(-5,0),(5,0) ;$ vertices: $(-8,0),(8,0)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Foci: $(-2,0),(2,0) ;$ vertices: $(-6,0),(6,0)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Foci $(0,-4),(0,4) ;$ vertices: $(0,-7),(0,7)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Foci: $(0,-3),(0,3) ;$ vertices: $(0,-4),(0,4)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Foci: $(-2,0),(2,0) ; y$ -intercepts: $-3$ and 3
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Foci: $(0,-2),(0,2) ; x$ -intercepts: $-2$ and 2
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Major axis horizontal with length $8 ;$ length of minor axis $=4$ center; $(0,0)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Major axis horizontal with length $12 ;$ length of minor axis $=6$ center: $(0,0)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Major axis vertical with length $10 ;$ length of minor axis $=4$ center: $(-2,3)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Major axis vertical with length $20 ;$ length of minor axis $=10$ center: $(2,-3)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Endpoints of major axis: $(7,9)$ and $(7,3)$ Endpoints of minor axis: $(5,6)$ and $(9,6)$
In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.Endpoints of major axis: $(2,2)$ and $(8,2)$ Endpoints of minor axis: $(5,3)$ and $(5,1)$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$(x+3)^{2}+4(y-2)^{2}=16$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$(x-3)^{2}+9(y+2)^{2}=18$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x-4)^{2}}{9}+\frac{(y+2)^{2}}{25}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x-3)^{2}}{9}+\frac{(y+1)^{2}}{16}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{x^{2}}{25}+\frac{(y-2)^{2}}{36}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x-4)^{2}}{4}+\frac{y^{2}}{25}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x+3)^{2}}{9}+(y-2)^{2}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x+2)^{2}}{16}+(y-3)^{2}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x-1)^{2}}{2}+\frac{(y+3)^{2}}{5}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$\frac{(x+1)^{2}}{2}+\frac{(y-3)^{2}}{5}=1$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$9(x-1)^{2}+4(y+3)^{2}=36$$
In Exercises $37-50,$ graph each ellipse and give the location of its foci.$$36(x+4)^{2}+(y+3)^{2}=36$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$9 x^{2}+25 y^{2}-36 x+50 y-164=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$4 x^{2}+9 y^{2}-32 x+36 y+64=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$9 x^{2}+16 y^{2}-18 x+64 y-71=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$x^{2}+4 y^{2}+10 x-8 y+13=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$4 x^{2}+y^{2}+16 x-6 y-39=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$4 x^{2}+25 y^{2}-24 x+100 y+36=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$49 x^{2}+16 y^{2}+98 x-64 y-671=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$36 x^{2}+9 y^{2}-216 x=0$$
In Exercises $51-60,$ convert each equation to standard form by completing the square on $x$ and $y .$ Then graph the ellipse and give the location of its foci.$$16 x^{2}+25 y^{2}-300 y+500=0$$
In Exercises $61-66,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\{\begin{array}{l}{x^{2}+y^{2}=1} \\{x^{2}+9 y^{2}=9}\end{array}\right.$$
In Exercises $61-66,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\{\begin{array}{r}{x^{2}+y^{2}=25} \\{25 x^{2}+y^{2}=25}\end{array}\right.$$
In Exercises $61-66,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\{\begin{array}{r}{\frac{x^{2}}{25}+\frac{y^{2}}{9}=1} \\{y=3}\end{array}\right.$$
In Exercises $61-66,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\{\begin{aligned}\frac{x^{2}}{4}+\frac{y^{2}}{36} &=1 \\x &=-2\end{aligned}\right.$$
In Exercises $61-66,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\{\begin{array}{c}{4 x^{2}+y^{2}=4} \\{2 x-y=2}\end{array}\right.$$
In Exercises $61-66,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\{\begin{aligned}4 x^{2}+y^{2} &=4 \\x+y &=3\end{aligned}\right.$$
In Exercises $67-68,$ graph each semiellipse.$$y=-\sqrt{16-4 x^{2}}$$
In Exercises $67-68,$ graph each semiellipse.$$y=-\sqrt{4-4 x^{2}}$$
Will a truck that is 8 feet wide carrying a load that reaches 7 feet above the ground clear the semielliptical arch on the one-way road that passes under the bridge shown in the figure?
A semielliptic archway has a height of 20 feet and a width of 50 feet, as shown in the figure. Can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane?
The elliptical ceiling in Statuary Hall in the U.S. Capitol Building is 96 feet long and 23 feet tall.a. Using the rectangular coordinate system in the figure shown, write the standard form of the equation of the elliptical ceiling.b. John Quincy Adams discovered that he could overhear the conversations of opposing party leaders near the left side of the chamber if he situated his desk at the focus at the right side of the chamber. How far from the center of the ellipse along the major axis did Adams situate his desk? (Round to the nearest foot.)
If an elliptical whispering room has a height of 30 feet and a width of 100 feet, where should two people stand if they would like to whisper back and forth and be heard?
What is an ellipse?
Describe how to graph $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$
Describe how to locate the foci for $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$
Describe one similarity and one difference between the graphs of $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$
Describe one similarity and one difference between the graphs of $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and $\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1$
An elliptipool is an elliptical pool table with only one pocket. A pool shark places a ball on the table, hits it in what appears to be a random direction, and yet it bounces off the edge, falling directly into the pocket. Explain why this happens.
Use a graphing utility to graph any five of the ellipses that you graphed by hand in Exercises $1-18$
Use a graphing utility to graph any three of the ellipses that you graphed by hand in Exercises $37-50 .$ First solve the given equation for $y$ by using the square root property. Enter each of the two resulting equations to produce each half of the ellipse.
Use a graphing utility to graph any one of the ellipses that you graphed by hand in Exercises $51-60 .$ Write the equation as a quadratic equation in $y$ and use the quadratic formula to solve for $y .$ Enter each of the two resulting equations to produce each half of the ellipse.
Write an equation for the path of each of the following elliptical orbits. Then use a graphing utility to graph the two ellipses in the same viewing rectangle. Can you see why early astronomers had difficulty detecting that these orbits are ellipses rather than circles?$\cdot$ Earth's orbit: Length of major axis: 186 million miles Length of minor axis: 185.8 million miles$\cdot$ Mars's orbit: Length of major axis: 283.5 million miles Length of minor axis: 278.5 million miles
Determine whether each statement makes sense or does not make sense, and explain your reasoning.I graphed an ellipse with a horizontal major axis and foci on the $y$ -axis.
Determine whether each statement makes sense or does not make sense, and explain your reasoning.I graphed an ellipse that was symmetric about its major axis but not symmetric about its minor axis.
Determine whether each statement makes sense or does not make sense, and explain your reasoning.You told me that an ellipse centered at the origin has vertices at $(-5,0)$ and $(5,0),$ so 1 was able to graph the ellipse.
Determine whether each statement makes sense or does not make sense, and explain your reasoning.In a whispering gallery at our science museum, I stood at one focus, my friend stood at the other focus, and we had a clear conversation, very little of which was heard by the 25 museum visitors standing between us.
Find the standard form of the equation of an ellipse with vertices at $(0,-6)$ and $(0,6),$ passing through $(2,-4)$
An Earth satellite has an elliptical orbit described by $$\frac{x^{2}}{(5000)^{2}}+\frac{y^{2}}{(4750)^{2}}=1$$(All units are in miles.) The coordinates of the center of Earth are $(16,0)$a. The perigee of the satellite's orbit is the point that is nearest Earth's center. If the radius of Earth is approximately 4000 miles, find the distance of the perigee above Earth's surface.b. The apogee of the satellite's orbit is the point that is the greatest distance from Earth's center. Find the distance of the apogee above Earth's surface.
The equation of the red ellipse in the figure shown is $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$ Write the equation for each circle shown in the figure.
What happens to the shape of the graph of $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ as $\frac{c}{a} \rightarrow 0,$ where $c^{2}=a^{2}-b^{2} ?$
Solve by eliminating variables: $$\left\{\begin{aligned}x-6 y &=-22 \\2 x+4 y-3 z &=29 \\3 x-2 y+5 z &=-17\end{aligned}\right.$$
Graph the solution set of the system:$$\left\{\begin{array}{l}{2 x+y \leq 4} \\{x>-3} \\{y \geq 1}\end{array}\right.$$
Where possible, find each product.a. $\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]$b. $\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{rrr}{-1} & {0} & {1} \\ {0} & {-1} & {1}\end{array}\right]$
Use the Law of sines to solve triangle $A B C$ if $A=35^{\circ}, a=11,$ and$b=15 .$ Assume $B$ is acute. Round lengths of sides to the nearest tenth and angle measures to the nearest.
Exercises $95-97$ will help you prepare for the material covered in the next section.Divide both sides of $4 x^{2}-9 y^{2}=36$ by 36 and simplify. How does the simplified equation differ from that of an ellipse?
Exercises $95-97$ will help you prepare for the material covered in the next section.Consider the equation $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$a. Find the $x$ -intercepts.b. Explain why there are no $y$ -intercepts.
Exercises $95-97$ will help you prepare for the material covered in the next section.Consider the equation $\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$a. Find the $y$ -intercepts.b. Explain why there are no $x$ -intercepts.