Chapter 10

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0)\). As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Sketch the reciprocal spiral given by \(r=c / \theta\). For \(c>0\), does it unwind in the clockwise direction?

Find the area of the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\).

Name the conic \(y^{2}=L x+K x^{2}\) according to the value of \(K\) and then show that in every case \(|L|\) is the length of the latus rectum of the conic. Assume that \(L \neq 0\).

Find the volume of the solid obtained by revolving the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\) about the \(y\)-axis.

Find the area of the region between the curve \(x=e^{2 t}, y=e^{-t}\), and the \(x\)-axis from \(t=0\) to \(t=\ln 5\). Make a sketch.

The graph of \(x \cos \alpha+y \sin \alpha=d\) is a line. Show that the perpendicular distance from the origin to this line is \(|d|\) by making a rotation of axes through the angle \(\alpha\).

The path of a projectile fired from level ground with a speed of \(v_{0}\) feet per second at an angle \(\alpha\) with the ground is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t, \quad y=-16 t^{2}+\left(v_{0} \sin \alpha\right) t $$ (a) Show that the path is a parabola. (b) Find the time of flight. (c) Show that the range (horizontal distance traveled) is \(\left(v_{0}^{2} / 32\right) \sin 2 \alpha\). (d) For a given \(v_{0}\), what value of \(\alpha\) gives the largest possible range?

Transform the equation \(x^{1 / 2}+y^{1 / 2}=a^{1 / 2}\) by a rotation of axes through \(45^{\circ}\) and then square twice to eliminate radicals on variables. Identify the corresponding curve.

Find the dimensions of the rectangle having the greatest possible area that can be inscribed in the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\). Assume that the sides of the rectangle are parallel to the axes of the ellipse.

Find the point in the first quadrant where the two hyperbolas \(25 x^{2}-9 y^{2}=225\) and \(-25 x^{2}+18 y^{2}=450\) intersect.

Find the points of \(x^{2}+14 x y+49 y^{2}=100\) that are closest to the origin.

Find the points of intersection of \(x^{2}+4 y^{2}=20\) and \(x+2 y=6\).

Recall that \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) transforms to \(a u^{2}+b u v+c v^{2}+d u+e v+f=0\) under a rotation of axes. Find formulas for \(a\) and \(c\), and show that \(a+c=A+C\).

If \(b=a\), the equations in Problem 63 are $$ \begin{aligned} &x=2 a \cos t-a \cos 2 t \\ &y=2 a \sin t-a \sin 2 t \end{aligned} $$ Find a Cartesian equation of the epicycloid by eliminating the parameter \(t\) between the equations.

A ball placed at a focus of an elliptical billiard table is shot with tremendous force so that it continues to bounce off the cushions indefinitely. Describe its ultimate path?

Consider the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\). (a) Show that its perimeter is $$ P=4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \cos ^{2} t} d t $$ where \(e\) is the eccentricity. C (b) The integral in part (a) is called an elliptic integral. It has been studied at great length, and it is known that the integrand does not have an elementary antiderivative, so we must turn to approximate methods to evaluate \(P\). Do so when \(a=1\) and \(e=\frac{1}{4}\) using the Parabolic Rule with \(n=4\). (Your answer should be near \(2 \pi\). Why?) (c) Repeat part (b) using \(n=20\).

Show that, if \(A+C\) and \(\Delta=4 A C-B^{2}\) are both positive, then the graph of \(A x^{2}+B x y+C y^{2}=1\) is an ellipse (or circle) with area \(2 \pi / \sqrt{\Delta}\). (Recall from Problem 55 of Section \(10.2\) that the area of the ellipse \(x^{2} / p^{2}+y^{2} / q^{2}=1\) is \(\pi p q\).)

The parametric curve given by \(x=\cos a t\) and \(y=\sin b t\) is known as a Lissajous figure. The \(x\)-coordinate oscillates \(a\) times between 1 and \(-1\) as \(t\) goes from 0 to \(2 \pi\), while the \(y\)-coordinate oscillates \(b\) times over the same \(t\) interval. This behavior is repeated over every interval of length \(2 \pi\). The entire motion takes place in a unit square. Plot the following Lissajous figures for a range of \(t\) that ensures that the resulting figure is a closed curve. In each case, count the number of times that the curve touches the horizontal and vertical borders of the unit square. (a) \(x=\sin t, y=\cos t\) (b) \(x=\sin 3 t, y=\cos 5 t\) (c) \(x=\cos 5 t, y=\sin 15 t\) (d) \(x=\sin 2 t, y=\cos 9 t\)

For what values of \(B\) is the graph of \(x^{2}+B x y+y^{2}=1\) (a) an ellipse (b) a circle (c) a hyperbola (d) two parallel lines

Plot the Lissajous figure defined by \(x=\cos 2 t\), \(y=\sin 7 t, 0 \leq t \leq 2 \pi\). Explain why this is a closed curve even though its graph does not look closed.

Listeners \(A(-8,0), B(8,0)\), and \(C(8,10)\) recorded the exact times at which they heard an explosion. If \(B\) and \(C\) heard the explosion at the same time and \(A\) heard it 12 seconds later, where was the explosion? Assume that distances are in kilometers and that sound travels \(\frac{1}{3}\) kilometer per second.

Show that \(\left(\sqrt{x^{2}-a^{2}}-x\right) \rightarrow 0\) as \(x \rightarrow \infty\).

Plot the following parametric curves. Describe in words how the point moves around the curve in each case. (a) \(x=\cos \left(t^{2}-t\right), y=\sin \left(t^{2}-t\right)\) (b) \(x=\cos \left(2 t^{2}+3 t+1\right), y=\sin \left(2 t^{2}+3 t+1\right)\) (c) \(x=\cos (-2 \ln t), y=\sin (-2 \ln t)\) (d) \(x=\cos (\sin t), y=\sin (\sin t)\)

Using a computer algebra system, plot the following parametric curves for \(0 \leq t \leq 2\). Describe the shape of the curve in each case and the similarities and differences among all the curves. (a) \(x=t, y=t^{2}\) (b) \(x=t^{3}, y=t^{6}\) (c) \(x=-t^{4}, y=-t^{8}\) (d) \(x=t^{5}, y=t^{10}\)

Plot the graph of the hypocycloid (see Problem 61) $$ \begin{aligned} &x=(a-b) \cos t+b \cos \frac{a-b}{b} t, \\ &y=(a-b) \sin t-b \sin \frac{a-b}{b} t \end{aligned} $$ for appropriate values of \(t\) in each of the following cases: (a) \(a=4, b=1\) (b) \(a=3, b=1\) (c) \(a=5, b=2\) (d) \(a=7, b=4\) Experiment with other positive integer values of \(a\) and \(b\) and then make conjectures about the length of the \(t\)-interval required for the curve to return to its starting point and about the number of cusps. What can you say if \(a / b\) is irrational?

Let \(P\) be a point on a ladder of length \(a+b, P\) being \(a\) units from the top end. As the ladder slides with its top end on the \(y\)-axis and its bottom end on the \(x\)-axis, \(P\) traces out a curve. Find the equation of this curve.

Show that a line through a focus of a hyperbola and perpendicular to an asymptote intersects that asymptote on the directrix nearest the focus.

Draw the Folium of Descartes \(x=3 t /\left(t^{3}+1\right)\), \(y=3 t^{2} /\left(t^{3}+1\right)\). Then determine the values of \(t\) for which this graph is in each of the four quadrants.

If a horizontal hyperbola and a vertical hyperbola have the same asymptotes, show that their eccentricities \(e\) and \(E\) satisfy \(e^{-2}+E^{-2}=1\).

Let \(C\) be the curve of intersection of a right circular cylinder and a plane making an angle \(\phi(0<\phi<\pi / 2)\) with the axis of the cylinder. Show that \(C\) is an ellipse.

Using the same axes, draw the conics \(y=\) \(\pm\left(a x^{2}+1\right)^{1 / 2}\) for \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\) using \(a=\) \(-2,-1,-0.5,-0.1,0,0.1,0.6,1\). Make a conjecture about how the shape of the figure depends on \(a\).