Chapter 12

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \(( \pm 1,0),\) asymptotes: \(y=\pm 5 x\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{7}{2-5 \sin \theta} $$

\(35-38\) Use a graphing device to graph the conic. $$ 2 x^{2}-4 x+y+5=0 $$

Solve the equations $$ \begin{aligned} x &=X \cos \phi-Y \sin \phi \\ y &=X \sin \phi+Y \cos \phi \end{aligned} $$ for \(X\) and \(Y\) in terms of \(x\) and \(y\) . [Hint: To begin, multiply the first equation by \(\cos \phi\) and the second by \(\sin \phi,\) and then add the two equations to solve for \(X . ]\)

Find an equation for the ellipse that satisfies the given conditions. Length of major axis: 4 , length of minor axis: \(2,\) foci on \(y\) -axis

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=-10\)

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \((0, \pm 6),\) asymptotes: \(y=\pm \frac{1}{3} X\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{8}{3+\cos \theta} $$

\(35-38\) Use a graphing device to graph the conic. $$ 4 x^{2}+9 y^{2}-36 y=0 $$

Show that the graph of the equation $$ \sqrt{x}+\sqrt{y}=1 $$ is part of a parabola by rotating the axes through an angle of \(45^{\circ} .[\text { Hint: First convert the equation to one that does not }\) involve radicals. \(]\)

Find an equation for the ellipse that satisfies the given conditions. Length of major axis: \(6,\) length of minor axis: \(4,\) foci on \(x\) -axis

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 8),\) asymptotes: \(y=\pm \frac{1}{2} x\)

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$ r=\frac{1}{4-3 \cos \theta} ; \quad \theta=\frac{\pi}{3} $$

\(35-38\) Use a graphing device to graph the conic. $$ 9 x^{2}+36=y^{2}+36 x+6 y $$

Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 2),\) length of minor axis: 6

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus on the positive \(x\) -axis, 2 units away from the directrix

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \((0, \pm 6),\) hyperbola passes through \((-5,9)\)

\(35-38\) Use a graphing device to graph the conic. $$ x^{2}-4 y^{2}+4 x+8 y=0 $$

Find an equation for the ellipse that satisfies the given conditions. Foci: \(( \pm 5,0),\) length of major axis: 12

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix has \(y\) -intercept 6

Find an equation for the hyperbola that satisfies the given conditions. Asymptotes: \(y=\pm x,\) hyperbola passes through \((5,3)\)

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$ r=\frac{2}{1+\sin \theta} ; \quad \theta=-\frac{\pi}{4} $$

Determine what the value of \(F\) must be if the graph of the equation $$4 x^{2}+y^{2}+4(x-2 y)+F=0$$ is (a) an ellipse, (b) a single point, or (c) the empty set.

Geometric lnvariants Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance \(d(P, Q)\) and \(d\left(P^{\prime}, Q^{\prime}\right)\) where \(P^{\prime}\) and \(Q^{\prime}\) are the images of \(P\) and \(Q\) under a rotation of axes.

Find an equation for the ellipse that satisfies the given conditions. Endpoints of major axis: \(( \pm 10,0),\) distance between foci: 6

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Opens upward with focus 5 units from the vertex

Find an equation for the hyperbola that satisfies the given conditions. Foci: \(( \pm 3,0),\) hyperbola passes through \((4,1)\)

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$ r=\frac{9}{2+2 \cos \theta} ; \quad \theta=-\frac{5 \pi}{6} $$

Find an equation for the ellipse that satisfies the given conditions. Endpoints of minor axis: \((0, \pm 3),\) distance between foc: 8

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focal diameter 8 and focus on the negative \(y\) -axis

Find an equation for the hyperbola that satisfies the given conditions. Foci: \(( \pm 5,0),\) length of transverse axis: 6

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6,0.8\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

This exercise deals with confocal parabolas, that is, families of parabolas that have the same focus. (a) Draw graphs of the family of parabolas $$x^{2}=4 p(y+p)$$ for \(p=-2,-\frac{3}{2},-1,-\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2\) (b) Show that each parabola in this family has its focus at the origin. (c) Describe the effect on the graph of moving the vertex closer to the origin.

Find an equation for the ellipse that satisfies the given conditions. Length of major axis: \(10,\) foci on \(x\) -axis, ellipse passes through the point \((\sqrt{5}, 2)\)

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 1),\) length of transverse axis: 1

(a) Graph the conics $$ r=\frac{e d}{(1+e \sin \theta)} $$ for \(e=1\) and various values of \(d\) . How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\) . How does the value of \(e\) affect the shape of the conic?

Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(\frac{1}{9},\) foci: \((0, \pm 2)\)

(a) Show that the asymptotes of the hyperbola \(x^{2}-y^{2}=5\) are perpendicular to each other. (b) Find an equation for the hyperbola with foci \(( \pm c, 0)\) and with asymptotes perpendicular to each other.

Orbit of a Satellite A satellite is in an elliptical orbit around the earth with the center of the earth at one focus, as shown in the figure at the top of the right-hand column. The height of the satellite above the earth varies between 140 \(\mathrm{mi}\) and 440 \(\mathrm{mi}\) . Assume that the earth is a sphere with radius 3960 \(\mathrm{mi}\) Find an equation for the path of the satellite with the origin at the center of the earth.

Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(0.8,\) foci: \(( \pm 1.5,0)\)

The hyperbolas $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { and } \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1 $$ are said to be conjugate to each other. (a) Show that the hyperbolas $$x^{2}-4 y^{2}+16=0 \quad\( and \)\quad 4 y^{2}-x^{2}+16=0$$ are conjugate to each other, and sketch their graphs on the same coordinate axes. (b) What do the hyperbolas of part (a) have in common? (c) Show that any pair of conjugate hyperbolas have the relationship you discovered in part (b).

Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(\sqrt{3} / 2,\) foci on \(y\) -axis, length of major axis: 4

In the derivation of the equation of the hyperbola at the beginning of this section, we said that the equation $$ \sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\pm 2 a $$ simplifies to $$ \left(c^{2}-a^{2}\right) x^{2}-a^{2} y^{2}=a^{2}\left(c^{2}-a^{2}\right) $$ Supply the steps needed to show this.

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. $$\left\\{\begin{array}{l}{4 x^{2}+y^{2}=4} \\ {4 x^{2}+9 y^{2}=36}\end{array}\right.$$

(a) For the hyperbola $$ \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 $$ determine the values of \(a, b,\) and \(c,\) and find the coordinates of the foci \(F_{1}\) and \(F_{2}\) . (b) Show that the point \(P\left(5, \frac{16}{3}\right)\) lies on this hyperbola. (c) Find \(d\left(P, F_{1}\right)\) and \(d\left(P, F_{2}\right) .\) (d) Verify that the difference between \(d\left(P, F_{1}\right)\) and \(d\left(P, F_{2}\right)\) is \(2 a .\)

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. $$ \left\\{\begin{array}{l}{\frac{x^{2}}{16}+\frac{y^{2}}{9}=1} \\\ {\frac{x^{2}}{9}+\frac{y^{2}}{16}=1}\end{array}\right. $$

Hyperbolas are called confocal if they have the same foci. (a) Show that the hyperbolas $$ \frac{y^{2}}{k}-\frac{x^{2}}{16-k}=1 \quad \text { with } 0

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. $$ \left\\{\begin{array}{c}{100 x^{2}+25 y^{2}=100} \\\ {x^{2}+\frac{y^{2}}{9}=1}\end{array}\right. $$

(a) Use a graphing device to sketch the top half (the portion in the first and second quadrants) of the family of ellipses \(X^{2}+k y^{2}=100\) for \(k=4,10,25,\) and \(50 .\) (b) What do the members of this family of ellipses have in common? How do they differ?

Ripples in Pool Two stones are dropped simultaneously into a calm pool of water. The crests of the resulting waves form equally spaced concentric circles, as shown in the figures. The waves interact with each other to create certain interference patterns. (a) Explain why the red dots lie on an ellipse. (b) Explain why the blue dots lie on a hyperbola.