Section 1
Conics and Calculus
Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)$$y^{2}=4 x$$
Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)$$(x+4)^{2}=-2(y-2)$$
Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)
Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)$$\frac{(x-2)^{2}}{16}+\frac{(y+1)^{2}}{4}=1$$
Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)$$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$
Match the equation with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).](GRAPH CANNOT COPY)$$\frac{(x-2)^{2}}{9}-\frac{y^{2}}{4}=1$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$y^{2}=-8 x$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$x^{2}+6 y=0$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$(x+5)+(y-3)^{2}=0$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$(x-6)^{2}+8(y+7)=0$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$y^{2}-4 y-4 x=0$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$y^{2}+6 y+8 x+25=0$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$x^{2}+4 x+4 y-4=0$$
Find the vertex, focus, and directrix of the parabola, and sketch its graph.$$y^{2}+4 y+8 x-12=0$$
Find an equation of the parabola.Vertex: (5,4)Focus: (3,4)
Find an equation of the parabola.Vertex: (-2,1)Focus: (-2,-1)
Find an equation of the parabola.Vertex: (0,5)Directrix: $y=-3$
Find an equation of the parabola.Focus: (2,2)Directrix: $x=-2$
Find an equation of the parabola.Vertex: (0,4)Points on the parabola:(-2,0),(2,0)
Find an equation of the parabola.Vertex: (2,4)Points on the parabola:(0,0),(4,0)
Find an equation of the parabola.Axis is parallel to $y$ -axis; graph passes through (0,3),(3,4) and (4,11)
Find an equation of the parabola.Directrix: $y=-2 ;$ endpoints of latus rectum are (0,2) and (8,2)
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.$$16 x^{2}+y^{2}=16$$
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.$$3 x^{2}+7 y^{2}=63$$
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.$$\frac{(x-3)^{2}}{16}+\frac{(y-1)^{2}}{25}=1$$
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.$$(x+4)^{2}+\frac{(y+6)^{2}}{1 / 4}=1$$
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.$$9 x^{2}+4 y^{2}+36 x-24 y+36=0$$
Find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.$$16 x^{2}+25 y^{2}-64 x+150 y+279=0$$
Find an equation of the ellipse.Center: (0,0)Focus: (5,0)Vertex: (6,0)
Find an equation of the ellipse.Vertices: (0,3),(8,3)Eccentricity: $\frac{3}{4}$
Find an equation of the ellipse.Vertices: (3,1),(3,9)Minor axis length: 6
Find an equation of the ellipse.Foci: (0,Ā±9)Major axis length: 22
Find an equation of the ellipse.Center: (0,0)Major axis: horizontalPoints on the ellipse:(3,1),(4,0)
Find an equation of the ellipse.Center: (1,2)Major axis: verticalPoints on the ellipse:(1,6),(3,2)
Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.$$\frac{x^{2}}{25}-\frac{y^{2}}{16}=1$$
Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.$$\frac{(y+3)^{2}}{225}-\frac{(x-5)^{2}}{64}=1$$
Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.$$9 x^{2}-y^{2}-36 x-6 y+18=0$$
Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.$$y^{2}-16 x^{2}+64 x-208=0$$
Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.$$x^{2}-9 y^{2}+2 x-54 y-80=0$$
Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.$$9 x^{2}-4 y^{2}+54 x+8 y+78=0$$
Find an equation of the hyperbola.Vertices: (Ā±1,0)Asymptotes: $y=\pm 5 x$
Find an equation of the hyperbola.Vertices: (0,Ā±4)Asymptotes: $y=\pm 2 x$
Find an equation of the hyperbola.Vertices: (2,Ā±3)Point on graph: (0,5)
Find an equation of the hyperbola.Vertices: (2,Ā±3)Foci: (2,Ā±5)
Find an equation of the hyperbola.Center: (0,0)Vertex: (0,2)Focus: (0,4)
Find an equation of the hyperbola.Center: (0,0)Vertex: (6,0)Focus: (10,0)
Find an equation of the hyperbola.Vertices: (0,2),(6,2)Asymptotes: $y=\frac{2}{3} x$$$y=4-\frac{2}{3} x$$
Find an equation of the hyperbola.Focus: (20,0)Asymptotes: $y=\pm \frac{3}{4} x$
Find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of $x.$$$\frac{x^{2}}{9}-y^{2}=1, \quad x=6$$
Find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of $x.$$$\frac{y^{2}}{4}-\frac{x^{2}}{2}=1, x=4$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$x^{2}+4 y^{2}-6 x+16 y+21=0$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$4 x^{2}-y^{2}-4 x-3=0$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$25 x^{2}-10 x-200 y-119=0$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$y^{2}-4 y=x+5$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$9 x^{2}+9 y^{2}-36 x+6 y+34=0$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$2 x(x-y)=y(3-y-2 x)$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$3(x-1)^{2}=6+2(y+1)^{2}$$
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.$$9(x+3)^{2}=36-4(y-2)^{2}$$
(a) Give the definition of a parabola.(b) Give the standard forms of a parabola with vertex at $(h, k)$(c) In your own words, state the reflective property of a parabola.
(a) Give the definition of an ellipse.(b) Give the standard form of an ellipse with center at $(h, k)$
(a) Give the definition of a hyperbola.(b) Give the standard forms of a hyperbola with center at $(h, k)$(c) Write equations for the asymptotes of a hyperbola.
Define the eccentricity of an ellipse. In your own words, describe how changes in the eccentricity affect the ellipse.
Consider the equation $9 x^{2}+4 y^{2}-36 x-24 y-36=0$(a) Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.(b) Change the $4 y^{2}$ -term in the equation to $-4 y^{2}$. Classify the graph of the new equation.(c) Change the $9 x^{2}$ -term in the original equation to $4 x^{2}$ Classify the graph of the new equation.(d) Describe one way you could change the original equation so that its graph is a parabola.
In parts (a)-(d), describe in words how a plane could intersect with the double-napped cone to form the conic section (see figure). (FIGURE CANNOT COPY)(a) Circle(b) Ellipse(c) Parabola(d) Hyperbola
A solar collector for heating water is constructed with a sheet of stainless steel that is formed into the shape of a parabola (see figure). The water will flow through a pipe that is located at the focus of the parabola. At what distance from the vertex is the pipe? (IMAGE CANNOT COPY)
A simply supported beam that is 16 meters long has a load concentrated at the center (see figure). The deflection of the beam at its center is 3 centimeters. Assume that the shape of the deflected beam is parabolic.(a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.)(b) How far from the center of the beam is the deflection 1 centimeter?
(a) Prove that any two distinct tangent lines to a parabola intersect.(b) Demonstrate the result of part (a) by finding the point of intersection of the tangent lines to the parabola $x^{2}-4 x-4 y=0$ at the points (0,0) and (6,3)
(a) Prove that if any two tangent lines to a parabola intersect at right angles, their point of intersection must lie on the directrix.(b) Demonstrate the result of part (a) by showing that the tangent lines to the parabola $x^{2}-4 x-4 y+8=0$ at the points (-2,5) and $\left(3, \frac{5}{4}\right)$ intersect at right angles, and that the point of intersection lies on the directrix.
Sketch the graphs of $x^{2}=4 p y$ for $p=\frac{1}{4}, \frac{1}{2}$ $1, \frac{3}{2},$ and 2 on the same coordinate axes. Discuss the change in the graphs as $p$ increases.
A cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 120 meters apart and 20 meters above the roadway (see figure). The cable touches the roadway midway between the towers. (IMAGE CANNOT COPY)(a) Find an equation for the parabolic shape of the cable.(b) Find the length of the parabolic cable.
A church window is bounded above by a parabola and below by the arc of a circle (see figure). Find the surface area of the window. (IMAGE CANNOT COPY)
A satellite signal receiving dish is formed by revolving the parabola given by $x^{2}=20 y$ about the $y$ -axis. The radius of the dish is $r$ feet. Verify that the surface area of the dish is given by $$2 \pi \int_{0}^{r} x \sqrt{1+\left(\frac{x}{10}\right)^{2}} d x=\frac{\pi}{15}\left[\left(100+r^{2}\right)^{3 / 2}-1000\right]$$
Earth moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is 149,598,000 kilometers, and the eccentricity is $0.0167 .$ Find the minimum distance (perihelion) and the maximum distance (aphelion) of Earth from the sun.
Satellite Orbit The apogee (the point in orbit farthest from Earth) and the perigee (the point in orbit closest to Earth) of an elliptical orbit of an Earth satellite are given by $A$ and $P$ Show that the eccentricity of the orbit is $$e=\frac{A-P}{A+P}$$
On November $27,1963,$ the United States launched the research satellite Explorer $18 .$ Its low and high points above the surface of Earth were 119 miles and 123,000 miles. Find the eccentricity of its elliptical orbit.
On November $20,1975,$ the United States launched the research satellite Explorer $55 .$ Its low and high points above the surface of Earth were 96 miles and 1865 miles. Find the eccentricity of its elliptical orbit.
Probably the most famous of all comets, Halley's comet, has an elliptical orbit with the sun at one focus. Its maximum distance from the sun is approximately $35.29 \mathrm{AU}$ (1 astronomical unit is approximately $\left.92.956 \times 10^{6} \text { miles }\right)$and its minimum distance is approximately $0.59 \mathrm{AU} .$ Find the eccentricity of the orbit.
Consider a particle traveling clockwise on the elliptical path $$\frac{x^{2}}{100}+\frac{y^{2}}{25}=1$$ The particle leaves the orbit at the point (-8,3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the $y$ -axis?
Find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid).$$\frac{x^{2}}{4}+\frac{y^{2}}{1}=1$$
Find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid).$$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$$
Use the integration capabilities of a graphing utility to approximate to two-decimal-place accuracy the elliptical integral representing the circumference of the ellipse$$\frac{x^{2}}{25}+\frac{y^{2}}{49}=1$$
(a) Show that the equation of an ellipse can be written as $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{a^{2}\left(1-e^{2}\right)}=1$(b) Use a graphing utility to graph the ellipse $\frac{(x-2)^{2}}{4}+\frac{(y-3)^{2}}{4\left(1-e^{2}\right)}=1$ for $e=0.95, e=0.75, e=0.5, e=0.25,$ and $e=0$(c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as $e$ approaches $0 .$
The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis? (FIGURE CANNOT COPY)
Prove Theorem 10.4 by showing that the tangent line to an ellipse at a point $P$ makes equal angles with lines through $P$ and the foci (see figure). [Hint: (1) Find the slope of the tangent line at $P,(2)$ find the slopes of the lines through $P$ and each focus, and ( 3 ) use the formula for the tangent of the angle between two lines.]
Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points (2,2) and (10,2) is 6.
Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.$
LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light $(186,000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on a rectangular coordinate system at (-150,0) and (150,0) and that a ship is traveling on a path with coordinates $(x, 75)$ (see figure). Find the $x$ -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds $(0.001 \text { second })$(FIGURE CANNOT COPY)
A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The mirror in the figure has the equation $\left(x^{2} / 36\right)-\left(y^{2} / 64\right)=1 .$ At which point on the mirror will light from the point (0,10) be reflected to the other focus?
Show that the equation of the tangent line to $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ at the point $$\left(x_{0}, y_{0}\right)$$ is $$\left(\frac{x_{0}}{a^{2}}\right) x-\left(\frac{y_{0}}{b^{2}}\right) y=1. $$
Prove that the graph of the equation $$A x^{2}+C y^{2}+D x+E y+F=0$$ is one of the following (except in degenerate cases).Conic(a) Circle(b) Parabola(c) Ellipse(d) Hyperbola Condition$A=C$$A=0$ or $C=0$ (but not both)$A C>0$$A C<0$
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.It is possible for a parabola to intersect its directrix.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The point on a parabola closest to its focus is its vertex.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If $C$ is the circumference of the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad b<a$$ then $2 \pi b \leq C \leq 2 \pi a$
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If $D \neq 0$ or $E \neq 0,$ then the graph of $y^{2}-x^{2}+D x+E y=0$ is a hyperbola.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the asymptotes of the hyperbola $\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1$ intersect at right angles, then $a=b$
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Every tangent line to a hyperbola intersects the hyperbola only at the point of tangency.
For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P$ Prove that $\left(P F_{1}\right)\left(P F_{2}\right) d^{2}$ is constant as $P$ varies on the ellipse, where $P F_{1}$ and $P F_{2}$ are the distances from $P$ to the foci $F_{1}$ and $F_{2}$ of the ellipse.
Find the minimum value of$(u-v)^{2}+\left(\sqrt{2-u^{2}}-\frac{9}{v}\right)^{2}$for $0<u<\sqrt{2}$ and $v>0$