Chapter 10

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (2,1)\(;\) axis \(y=1 ;(5,0)\) on graph.

(a) Find a parameterization of the line segment joining (-5,-3) and \((7,4),\) as in Exercises \(45-47\) (b) Explain why another parameterization of this line segment is given by $$\begin{array}{l}x=-5+12 \sin t \quad \text { and } \\\y=-3+7 \sin t \quad(0 \leq t \leq \pi / 2)\end{array}$$ (c) Use the trace feature to verify that the segment is traced out twice when the \(t\) -range in part (b) is changed to \(0 \leq t \leq \pi(\text { use } t-\text { step }=\pi / 20) .\) Explain why. (d) What happens when \(0 \leq t \leq 2 \pi ?\)

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (1,-3)\(;\) axis \(y=-3 ;(-1,-4)\) on graph.

(a) Graph the curve given by \(x=\sin k t \quad\) and \(\quad y=\cos t \quad(0 \leq t \leq 2 \pi)\) when \(k=1,2,3,\) and \(4 .\) Use the window with \(-1.5 \leq x \leq 1.5 \quad\) and \(\quad-1.5 \leq y \leq 1.5\) and \(t\) -step \(=\pi / 30\) (b) Without graphing, predict the shape of the graph when \(k=5\) and \(k=6 .\) Then verify your predictions graphically.

Find the equation of the ellipse that satisfies the given conditions. Center (3,-2)\(;\) passing through (3,-6) and (9,-2).

Find a rectangular equation that is equivalent to the given polar equation. $$r=2 \sin \theta$$

A comet travels in a parabolic orbit with the sun as focus. When the comet is 60 million miles from the sun, the line segment from the sun to the comet makes an angle of \(\pi / 3\) radians with the axis of the parabolic orbit. Using the sun as the pole and assuming the axis of the orbit lies along the polar axis, find a polar equation for the orbit.

Sketch the graph of \(\frac{y^{2}}{4}-\frac{x^{2}}{b^{2}}=1\) for \(b=2, b=4, b=8\) \(b=12,\) and \(b=20 .\) What happens to the hyperbola as \(b\) takes larger and larger values? Could the graph ever degenerate into a pair of horizontal lines?

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. $$\text { Vertex }(-3,-2) ; \text { focus }(-47 / 16,-2)$$

(a) Graph the curve given by \(x=3 \sin 2 t \quad\) and \(\quad y=2 \cos k t \quad(0 \leq t \leq 2 \pi)\) when \(k=1,2,3,4 .\) Use the window with \(-3.5 \leq x \leq\) 3.5 and \(-2.5 \leq y \leq 2.5\) and \(t\) -step \(=\pi / 30\) (b) Predict the shape of the graph when \(k=5,6,7,8 .\) Verify your predictions graphically.

Find a rectangular equation that is equivalent to the given polar equation. $$r=3 \cos \theta$$

Halley's Comet has an elliptical orbit, with eccentricity .97 and the sun as a focus. The length of the major axis of the orbit is 3364.74 million miles. Using the sun as the pole and assuming the major axis of the orbit is perpedicular to the polar axis, find a polar equation for the orbit.

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. $$\text { Vertex }(-5,-5) ; \text { focus }(-5,-99 / 20)$$

Find a number \(k\) such that (-2,1) is on the graph of \(3 x^{2}+k y^{2}=4 .\) Then graph the equation.

Find the equations of two distinct ellipses satisfying the given conditions. Center at (-5,3)\(;\) major axis of length \(14 ;\) minor axis of length 8 .

Find a rectangular equation that is equivalent to the given polar equation. $$r=\frac{4}{1+\sin \theta}$$

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (1,1)\(;\) focus \((1,9 / 8)\)

Show that the asymptotes of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{a^{2}}=1\) are perpendicular to each other.

Find the equations of two distinct ellipses satisfying the given conditions. Center at (2,-6)\(;\) major axis of length \(15 ;\) minor axis of length 6.

Find a rectangular equation that is equivalent to the given polar equation. $$r=\frac{6}{1-\cos \theta}$$

Find the approximate coordinates of the points where these hyperbolas intersect: $$ \frac{(x-1)^{2}}{4}-\frac{(y+1)^{2}}{8}=1 \quad \text { and } \quad 4 y^{2}-x^{2}=1 $$

Locate all local maxima and minima (other than endpoints of the curve. $$x=4 t-6, \quad y=3 t^{2}+2, \quad-10 \leq t \leq 10$$

Two listening stations that are 1 mile apart record an explosion. One microphone receives the sound 2 seconds after the other does. Use the line through the microphones as the \(x\) -axis, with the origin midway between the microphones, and the fact that sound travels at 1100 feet per second to find the equation of a hyperbola on which the explosion is located. Can you determine the exact location of the explosion?

Two transmission stations \(P\) and \(Q\) are located 200 miles apart on a straight shoreline. A ship 50 miles from shore is moving parallel to the shoreline. A signal from \(Q\) reaches the ship 400 microseconds after a signal from \(P .\) If the signals travel at 980 feet per microsecond, find the location of the ship (in terms of miles) in the coordinate system with \(x\) -axis through \(P\) and \(Q\) and origin midway between them.

Locate all local maxima and minima (other than endpoints of the curve. $$x=4 t^{3}-t+4, \quad y=-3 t^{2}+5, \quad-2 \leq t \leq 2$$

Sketch the graph of the equation without using a calculator. $$\theta=-\pi / 3$$

Show that the ball's path in Example 9 is a parabola by eliminating the parameter in the parametric equations \(x=\left(140 \cos 31^{\circ}\right) t \quad\) and \(\quad y=\left(140 \sin 31^{\circ}\right) t-16 t^{2}\) [Hint: Solve the first equation for \(t\), and substitute the result in the second equation. \(]\)

Use a calculator in degree mode and assume that air resistance is negligible. A skeet is fired from the ground with an initial velocity of 110 feet per second at an angle of \(28^{\circ}\) (a) Graph the skeet's path. (b) How long is the skeet in the air? (c) How high does it go?

Sketch the graph of the equation without using a calculator. $$\theta=1$$

If \(a>0\) and \(b>0,\) then the eccentricity of the hyperbola $$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}+b^{2}}}{a} .\) Find the eccentricity of the hyperbola whose equation is given. $$\frac{(x-6)^{2}}{10}-\frac{y^{2}}{40}=1$$

Use a calculator in degree mode and assume that air resistance is negligible. A ball is thrown from a height of 5 feet above the ground with an initial velocity of 60 feet per second at an angle of \(50^{\circ}\) with the horizontal. (a) Graph the ball's path. (b) When and where does the ball hit ground?

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

Use a calculator in degree mode and assume that air resistance is negligible. A medieval bowman shoots an arrow which leaves the bow 4 feet above the ground with an initial velocity of 88 feet per second at an angle of \(48^{\circ}\) with the horizontal. (a) Graph the arrow's path. (b) Will the arrow go over the 40 -foot-high castle wall that is 200 feet from the archer?

Sketch the graph of the equation. $$r=\theta \quad(\theta \leq 0)$$

If \(a>0\) and \(b>0,\) then the eccentricity of the hyperbola $$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}+b^{2}}}{a} .\) Find the eccentricity of the hyperbola whose equation is given. $$6(y-2)^{2}=18+3(x+2)^{2}$$

In Exercises \(55-62,\) identify the conic section whose equation is given, list its vertex or vertices, if any, and find its graph. $$3 x^{2}+3 y^{2}-6 x-12 y-6=0$$

Use a calculator in degree mode and assume that air resistance is negligible. A golfer at a driving range stands on a platform 2 feet above the ground and hits the ball with an initial velocity of 120 feet per second at an angle of \(39^{\circ}\) with the horizontal. There is a 32 -foot-high fence 400 feet away. Will the ball fall short, hit the fence, or go over it?

If \(a>0\) and \(b>0,\) then the eccentricity of the hyperbola $$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}+b^{2}}}{a} .\) Find the eccentricity of the hyperbola whose equation is given. $$16 x^{2}-9 y^{2}-32 x+36 y+124=0$$

In Exercises \(55-62,\) identify the conic section whose equation is given, list its vertex or vertices, if any, and find its graph. $$2 x^{2}+3 y^{2}+12 x-6 y+9=0$$

Use a calculator in degree mode and assume that air resistance is negligible. A golf ball is hit off the tee at an angle of \(30^{\circ}\) and lands 300 feet away. What was its initial velocity? [Hint: The ball lands when \(x=300\) and \(y=0 .\) Use this fact and the parametric equations for the ball's path to find two equations in the variables \(t \text { and } v . \text { Solve for } v .]\)

If \(a>0\) and \(b>0,\) then the eccentricity of the hyperbola $$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1$$ is the number \(\frac{\sqrt{a^{2}+b^{2}}}{a} .\) Find the eccentricity of the hyperbola whose equation is given. $$4 x^{2}-5 y^{2}-16 x-50 y+71=0$$

In Exercises \(55-62,\) identify the conic section whose equation is given, list its vertex or vertices, if any, and find its graph. $$2 x^{2}-y^{2}+16 x+4 y+24=0$$

Use a calculator in degree mode and assume that air resistance is negligible. A football kicked from the ground has an initial velocity of 75 feet per second. (a) Set up the parametric equations that describe the ball's path. Experiment graphically with different angles to find the smallest angle (within one degree) needed so that the ball travels at least 150 feet. (b) Use algebra and trigonometry to find the angle needed for the ball to travel exactly 150 feet. \([\text {Hint:}\) The ball lands when \(x=150\) and \(y=0 .\) Use this fact and the

Halley's Comet has an elliptical orbit with the sun as one focus and a major axis that is 1,636,484,848 miles long. The closest the comet comes to the sun is 54,004,000 miles. What is the maximum distance from the comet to the sun?

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

(a) Graph these hyperbolas (on the same screen if possible \()\) $$\frac{y^{2}}{4}-\frac{x^{2}}{1}=1 \quad \frac{y^{2}}{4}-\frac{x^{2}}{12}=1 \quad \frac{y^{2}}{4}-\frac{x^{2}}{96}=1$$ (b) Compute the eccentricity of each hyperbola in part (a). (c) On the basis of parts (a) and (b), how is the shape of a hyperbola related to its eccentricity?

Use a calculator in degree mode and assume that air resistance is negligible. A golf ball is hit off the ground at an angle of \(\theta\) degrees with an initial velocity of 100 feet per second. (a) Graph the path of the ball when \(\theta=20^{\circ}, \theta=40^{\circ}\) \(\theta=60^{\circ},\) and \(\theta=80^{\circ}\) (b) For what angle in part (a) does the ball land farthest from where it started? (c) Experiment with different angles, as in parts (a) and (b), and make a conjecture as to which angle results in the ball landing farthest from its starting point.

Use a calculator in degree mode and assume that air resistance is negligible. A golf ball is hit off the ground at an angle of \(\theta\) degrees with an initial velocity of 100 feet per second. (a) Graph the path of the ball when \(\theta=30^{\circ}\) and when \(\theta=60^{\circ} .\) In which case does the ball land farthest away? (b) Do part (a) when \(\theta=25^{\circ}\) and \(\theta=65^{\circ}\) (c) Experiment further, and make a conjecture as to the results when the sum of the two angles is \(90^{\circ} .\) (d) Prove your conjecture algebraically. [Hint: Find the value of \(t\) at which a ball hit at angle \(\theta\) hits the ground (which occurs when \(y=0\) ); this value of \(t\) will be an expression involving \(\theta .\) Find the corresponding value of \(x\) (which is the distance of the ball from the starting point). Then do the same for an angle of \(90^{\circ}-\theta\) and use the cofunction identities (in degrees) to show that you get the same value of \(x .]\)

Sketch the graph of the equation. $$r=-6 \sin \theta$$

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