Chapter 11

An angle of rotation is specified, followed by the coordinates of a point in the \(x^{\prime}-y^{\prime}\) system. Find the coordinates of each point with respect to the \(x\) -y system. $$\theta=30^{\circ} ;\left(x^{\prime}, y^{\prime}\right)=(\sqrt{3}, 2)$$

Graph each ellipse. Specify the eccentricity, the center, and the endpoints of the major and minor axes. (a) \(r=\frac{6}{3+2 \cos \theta}\) (b) \(r=\frac{6}{3-2 \cos \theta}\)

You are given an ellipse and a point P on the ellipse. Find \(F_{1} P\) and \(F_{2} P\), the lengths of the focal radii. $$x^{2}+3 y^{2}=76 ; P(-8,2)$$

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$x^{2}-4 y^{2}=4$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$4 x^{2}+9 y^{2}=36$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$x^{2}=4 y$$

To solve these problems, you will need to utilize the formulas listed at the beginning of this section. Find the distance between the points (-5,-6) and (3,-1).

An angle of rotation is specified, followed by the coordinates of a point in the \(x^{\prime}-y^{\prime}\) system. Find the coordinates of each point with respect to the \(x\) -y system. $$\theta=60^{\circ} ;\left(x^{\prime}, y^{\prime}\right)=(-1,1)$$

You are given an ellipse and a point P on the ellipse. Find \(F_{1} P\) and \(F_{2} P\), the lengths of the focal radii. $$x^{2}+3 y^{2}=57 ; P(3,-4)$$

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$y^{2}-x^{2}=1$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$4 x^{2}+25 y^{2}=100$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$x^{2}=16 y$$

An angle of rotation is specified, followed by the coordinates of a point in the \(x^{\prime}-y^{\prime}\) system. Find the coordinates of each point with respect to the \(x\) -y system. $$\theta=45^{\circ} ;\left(x^{\prime}, y^{\prime}\right)=(\sqrt{2},-\sqrt{2})$$

You are given an ellipse and a point P on the ellipse. Find \(F_{1} P\) and \(F_{2} P\), the lengths of the focal radii. $$\left(x^{2} / 15^{2}\right)+\left(y^{2} / 5^{2}\right)=1 ; P(9,4)$$

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$y^{2}-4 x^{2}=4$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$x^{2}+16 y^{2}=16$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$y^{2}=-8 x$$

An angle of rotation is specified, followed by the coordinates of a point in the \(x\) -y system. Find the coordinates of each point with respect to the \(x\) '-y' system. $$\theta=45^{\circ} ;(x, y)=(0,-2)$$

You are given an ellipse and a point P on the ellipse. Find \(F_{1} P\) and \(F_{2} P\), the lengths of the focal radii. $$2 x^{2}+3 y^{2}=14 ; P(-1,-2)$$

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$25 x^{2}-9 y^{2}=225$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$9 x^{2}+25 y^{2}=225$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$y^{2}=12 x$$

To solve these problems, you will need to utilize the formulas listed at the beginning of this section. Find an equation of the line passing through the points (6,3) and \((1,0) .\) Write your answer in the form \(y=m x+b\).

Graph each hyperbola. Specify the eccentricity, the center, and the values of \(a, b,\) and \(c .\) (a) \(r=\frac{3}{2+4 \cos \theta}\) (b) \(r=\frac{3}{2-4 \cos \theta}\)

Determine the foci, the eccentricity, and the directrices for each ellipse and hyperbola. (a) \(\left(x^{2} / 4^{2}\right)+\left(y^{2} / 3^{2}\right)=1\) (b) \(\left(x^{2} / 4^{2}\right)-\left(y^{2} / 3^{2}\right)=1\)

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$16 x^{2}-25 y^{2}=400$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$x^{2}+2 y^{2}=2$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$x^{2}=-20 y$$

To solve these problems, you will need to utilize the formulas listed at the beginning of this section. Find an equation of the line that is the perpendicular bisector of the line segment joining the points (2,1) and (6,7). Write your answer in the form \(A x+B y+C=0\).

An angle of rotation is specified, followed by the coordinates of a point in the \(x\) -y system. Find the coordinates of each point with respect to the \(x\) '-y' system. $$\theta=15^{\circ} ;(x, y)=(1,0)$$

Graph each hyperbola. Specify the eccentricity, the center, and the values of \(a, b,\) and \(c .\) (a) \(r=\frac{3}{3+4 \sin \theta}\) (b) \(r=\frac{3}{3-4 \sin \theta}\)

Determine the foci, the eccentricity, and the directrices for each ellipse and hyperbola. (a) \(x^{2}+4 y^{2}=1\) (b) \(x^{2}-4 y^{2}=1\)

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$9 x^{2}-y^{2}=36$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$2 x^{2}+3 y^{2}=3$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$x^{2}-y=0$$

Find \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is the (acute) angle of rotation that eliminates the \(x^{\prime} y^{\prime}\) -term. Note: You are not asked to graph the equation. $$25 x^{2}-24 x y+18 y^{2}+1=0$$

Graph each conic section. If the conic is a parabola, specify (using rectangular coordinates) the vertex and the directrix. If the conic is an ellipse, specify the center, the eccentricity, and the lengths of the major and minor axes. If the conic is a hyperbola, specify the center, the eccentricity, and the lengths of the transverse and conjugate axes. $$r=\frac{24}{2-3 \cos \theta}$$

Determine the foci, the eccentricity, and the directrices for each ellipse and hyperbola. (a) \(12 x^{2}+13 y^{2}=156\) (b) \(12 x^{2}-13 y^{2}=156\)

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$2 y^{2}-3 x^{2}=1$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$16 x^{2}+9 y^{2}=144$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$y^{2}+28 x=0$$

To solve these problems, you will need to utilize the formulas listed at the beginning of this section. Find the \(x\) - and \(y\) -intercepts of the circle with center (1,0) and radius 5.

Find \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is the (acute) angle of rotation that eliminates the \(x^{\prime} y^{\prime}\) -term. Note: You are not asked to graph the equation. $$x^{2}+24 x y+8 y^{2}-8=0$$

Graph each conic section. If the conic is a parabola, specify (using rectangular coordinates) the vertex and the directrix. If the conic is an ellipse, specify the center, the eccentricity, and the lengths of the major and minor axes. If the conic is a hyperbola, specify the center, the eccentricity, and the lengths of the transverse and conjugate axes. $$r=\frac{16}{10+5 \sin \theta}$$

Determine the foci, the eccentricity, and the directrices for each ellipse and hyperbola. (a) \(x^{2}+2 y^{2}=2\) (b) \(x^{2}-2 y^{2}=2\)

Graph the hyperbolas. In each case in which the hyperbola is nondegenerate, specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. also specify The centers. $$x^{2}-y^{2}=9$$

Graph the ellipses. In case, specify the lengths of the major and minor axes, the foci, and the eccentricity. For Exercises \(13-24,\) also specify the center of the ellipse. $$25 x^{2}+y^{2}=25$$

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$4 y^{2}+x=0$$

Find \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is the (acute) angle of rotation that eliminates the \(x^{\prime} y^{\prime}\) -term. Note: You are not asked to graph the equation. $$x^{2}-24 x y+8 y^{2}-8=0$$

Graph each conic section. If the conic is a parabola, specify (using rectangular coordinates) the vertex and the directrix. If the conic is an ellipse, specify the center, the eccentricity, and the lengths of the major and minor axes. If the conic is a hyperbola, specify the center, the eccentricity, and the lengths of the transverse and conjugate axes. $$r=\frac{8}{5+3 \sin \theta}$$