Chapter 10

If \(a>b>0,\) then the eccentricity of the ellipse $$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}-b^{2}}}{a} .\) Find the eccentricity of the ellipse whose equation is given. $$\frac{x^{2}}{100}+\frac{y^{2}}{99}=1$$

Sketch the graph of the equation. $$r=\sin 3 \theta$$

If \(a>b>0,\) then the eccentricity of the ellipse $$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}-b^{2}}}{a} .\) Find the eccentricity of the ellipse whose equation is given. $$\frac{x^{2}}{18}+\frac{y^{2}}{25}=1$$

Set your calculator for radian mode and for simultaneous graphing mode [check your instruction manual for how to do this]. Particles \(A, B,\) and \(C\) are moving in the plane, with their positions at time \(t\) seconds given by: \(A: \quad x=8 \cos t \quad\) and \(\quad y=5 \sin t\) \(B: \quad x=3 t \quad\) and \(\quad y=5 t\) \(\begin{array}{llll}C: & x=3 t & \text { and } & y=4 t\end{array}\) (a) Graph the paths of \(A\) and \(B\) in the window with \(0 \leq x \leq 12,0 \leq y \leq 6,\) and \(0 \leq t \leq 2 .\) The paths intersect, but do the particles actually collide? That is, are they at the same point at the same time? [For slow motion, choose a very small \(t \text { step, such as } .01 .]\) (b) Set \(t\) step \(=.05\) and use trace to estimate the time at which \(A\) and \(B\) are closest to each other. (c) Graph the paths of \(A\) and \(C\) and determine geometrically [as in part (b)] whether they collide. Approximately when are they closest? (d) Confirm your answers in part (c) as follows. Explain why the distance between particles \(A\) and \(C\) at time \(t\) is given by $$d=\sqrt{(8 \cos t-3 t)^{2}+(5 \sin t-4 t)^{2}}$$ \(A\) and \(C\) will collide if \(d=0\) at some time. Using function graphing mode, graph this distance function when \(0 \leq t \leq 2,\) and using zoom-in if necessary, show that \(d\) is always positive. Find the value of \(t\) for which \(d\) is smallest.

If \(a>b>0,\) then the eccentricity of the ellipse $$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}-b^{2}}}{a} .\) Find the eccentricity of the ellipse whose equation is given. $$\frac{(x-3)^{2}}{10}+\frac{(y-9)^{2}}{40}=1$$

Let \(P\) be a point at distance \(k\) from the center of a circle of radius \(r .\) As the circle rolls along the \(x\) -axis, \(P\) traces out a curve called a trochoid. [When \(k \leq r\), it might help to think of the circle as a bicycle wheel and \(P\) as a point on one of the spokes. \(]\) (a) Assume that \(P\) is on the \(y\) -axis as close as possible to the \(x\) -axis when \(t=0,\) and show that the parametric equations of the trochoid are $$x=r t-k \sin t \quad \text { and } \quad y=r-k \cos t$$ Note that when \(k=r,\) these are the equations of a cycloid. (b) Sketch the graph of the trochoid with \(r=3\) and \(k=2\) (c) Sketch the graph of the trochoid with \(r=3\) and \(k=4\)

If \(a>b>0,\) then the eccentricity of the ellipse $$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}-b^{2}}}{a} .\) Find the eccentricity of the ellipse whose equation is given. $$\frac{(x+5)^{2}}{12}+\frac{(y-4)^{2}}{8}=1$$

Sketch the graph of the equation. $$r=2+4 \cos \theta$$

Find the points of intersection of the parabola \(4 x^{2}-8 x=2 y+5\) and the line \(y=15\)

A satellite is to be placed in an elliptical orbit, with the center of the earth as one focus. The satellite's maximum distance from the surface of the earth is to be \(22,380 \mathrm{km},\) and its minimum distance is to be \(6540 \mathrm{km} .\) Assume that the radius of the earth is \(6400 \mathrm{km},\) and find the eccentricity of the satellite's orbit.

Sketch the graph of the equation. $$r=\sin \theta+\cos \theta$$

Let \(p\) be a real number. (a) Show that the endpoints of the latus rectum of the parabola with equation \(y^{2}=4 p x\) are \((p,-2 p)\) and \((p, 2 p)\) (b) Show that the endpoints of the latus rectum of the parabola with equation \(x^{2}=4 p y\) are \((-2 p, p)\) and \((2 p, p)\)

The first step in landing Apollo 11 on the moon was to place the spacecraft in an elliptical orbit such that the minimum distance from the surface of the moon to the spacecraft was \(110 \mathrm{km}\) and the maximum distance was \(314 \mathrm{km} .\) If the radius of the moon is \(1740 \mathrm{km},\) find the eccentricity of the Apollo 11 orbit.

Sketch the graph of the equation. $$r=4 \cos \theta+4 \sin \theta$$

Show that the length of the latus rectum of the parabola with equation \(\left.y^{2}=4 p x \text { or } x^{2}=4 p y \text { is } 4|p| . \text { [ Hint: Exercise } 73 .\right]\)

Consider the ellipse whose equation is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 .\) Show that if \(a=b,\) then the graph is actually a circle.

Sketch the graph of the equation. $$r=\sin (\theta / 2)$$

Complete the derivation of the equation of the ellipse on page 673 as follows. (a) By squaring both sides, show that the equation $$ \sqrt{(x+c)^{2}+y^{2}}=2 a-\sqrt{(x-c)^{2}+y^{2}} $$ may be simplified as $$a \sqrt{(x-c)^{2}+y^{2}}=a^{2}-c x.$$ (b) Show that the last equation in part (a) may be further simplified as $$\left(a^{2}-c^{2}\right) x^{2}+a^{2} y^{2}=a^{2}\left(a^{2}-c^{2}\right).$$

Sketch the graph of the equation. $$r=4 \tan \theta$$

Sketch the graph of the equation. $$r=\sin \theta \tan \theta \quad \text { (cissoid) }$$

A radio telescope has a parabolic dish with a diameter of 300 feet. Its receiver (focus) is located 130 feet from the vertex. How deep is the dish at its center? IHint: Position the dish as in Figure \(10-47,\) and find the equation of the parabola.

Sketch the graph of the equation. $$r^{2}=1 / \theta$$

A large spotlight has a parabolic reflector that is 3 feet deep at its center. The light source is located \(1 \frac{1}{3}\) feet from the vertex. What is the diameter of the reflector?

Sketch the graph of the equation. $$r=1 / \theta \quad(\theta > 0)$$

The cables of a suspension bridge are shaped like parabolas. The cables are attached to the towers 100 feet from the bridge surface, and the towers are 420 feet apart. The cables touch the bridge surface at the center (midway between the towers). At a point on the bridge 100 feet from one of the towers, how far is the cable from the bridge surface? (IMAGE CAN'T COPY)

At a point 120 feet from the center of a suspension bridge, the cables are 24 feet above the bridge surface. Assume that the cables are shaped like parabolas and touch the bridge surface at the center (which is midway between the towers). If the towers are 600 feet apart, how far above the surface of the bridge are the cables attached to the towers?

(a) Find a complete graph of \(r=1-2 \sin 3 \theta\) (b) Predict what the graph of \(r=1-2 \sin 4 \theta\) will look like. Then check your prediction with a calculator. (c) Predict what the graph of \(r=1-2 \sin 5 \theta\) will look like. Then check your prediction with a calculator.

(a) Find a complete graph of \(r=1-3 \sin 2 \theta\) (b) Predict what the graph of \(r=1-3 \sin 3 \theta\) will look like. Then check your prediction with a calculator. (c) Predict what the graph of \(r=1-3\) sin \(4 \theta\) will look like. Then check your prediction with a calculator.

If \(a, b\) are nonzero constants, show that the graph of \(r=\) \(a \sin \theta+b \cos \theta\) is a circle. \([\text {Hint}: \text { Multiply both sides by } r\) and convert to rectangular coordinates.]

Prove that the coordinate conversion formulas are valid when \(r < 0 .[\text {Hint: If } P \text { has coordinates }(x, y) \text { and }(r, \theta), \text { with } r < 0\) verify that the point \(Q\) with rectangular coordinates \((-x,-y)\) has polar coordinates \((-r, \theta) .\) since \(r < 0,-r\) is positive and the conversion formulas proved in the text apply to \(Q .\) For instance, \(-x=-r \cos \theta, \text { which implies that } x=r \cos \theta .]\)

Distance Formula for Polar Coordinates: Prove that the distance from \((r, \theta)\) to \((s, \beta)\) is $$ \sqrt{r^{2}+s^{2}-2 r s \cos (\theta-\beta)} $$ [Hint: If \(r > 0, s > 0,\) and \(\theta > \beta,\) then the triangle with vertices \((r, \theta),(s, \beta),(0,0)\) has an angle of \(\theta-\beta,\) whose sides have lengths \(r\) and \(s .\) Use the Law of cosines.]

Explain why the following symmetry tests for the graphs of polar equations are valid. (a) If replacing \(\theta\) by \(-\theta\) produces an equivalent equation, then the graph is symmetric with respect to the line \(\theta=0\) (the \(x\) -axis). (b) If replacing \(\theta\) by \(\pi-\theta\) produces an equivalent equation, then the graph is symmetric with respect to the line \(\theta=\pi / 2(\text { the } y\) -axis ) (c) If replacing \(r\) by \(-r\) produces an equivalent equation, then the graph is symmetric with respect to the origin.