Chapter 10

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=t^{3}\\\ &y=t^{2} \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \(\left(0, \frac{1}{2}\right)\)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$5 x^{2}-6 x y+5 y^{2}-12=0$$

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (±6,0)\(;\) passes through the point (4,1)

Identify the conic and sketch its graph. $$r=\frac{6}{1+\cos \theta}$$

Test for symmetry with respect to the line \(\theta=\pi 2,\) the polar axis, and the pole. $$r^{2}=25 \sin 2 \theta$$

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=-2$$

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=t^{3}\\\ &y=t^{4} \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \(\left(-\frac{3}{2}, 0\right)\)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$2 x^{2}+x y+2 y^{2}-8=0$$

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±5)\(;\) passes through the point (4,2)

Identify the conic and sketch its graph. $$r=\frac{2}{2-\cos \theta}$$

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(0, \pi)$$

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=1$$

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=t+1\\\ &y=\frac{t}{t+1} \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (-2,0)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}-16=0$$

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$x^{2}-y^{2}=1$$

Identify the conic and sketch its graph. $$r=\frac{4}{4+\sin \theta}$$

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(0,-\pi)$$

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=2$$

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=t-1\\\ &y=\frac{t}{t-1} \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (0,-2)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$7 x^{2}-6 \sqrt{3} x y+13 y^{2}-64=0$$

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$$

Identify the conic and sketch its graph. $$r=\frac{6}{2+\sin \theta}$$

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(3, \pi / 2)$$

Find the maximum value of \(|r|\) and any zeros of \(r\). $$r=4 \cos 3 \theta$$

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=\frac{3}{4}$$

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=2(t+1)\\\ &y=|t-2| \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=1\)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$x^{2}+2 x y+y^{2}+\sqrt{2} x-\sqrt{2} y=0$$

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{y^{2}}{25}-\frac{x^{2}}{81}=1$$

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (0,2),(8,2)\(;\) minor axis of length 2

Identify the conic and sketch its graph. $$r=\frac{9}{3-2 \cos \theta}$$

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(3,3 \pi / 2)$$

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=\frac{1}{2}$$

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=|t-1|\\\ &y=t+2 \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=-2\)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$3 x^{2}-2 \sqrt{3} x y+y^{2}+2 x+2 \sqrt{3} y=0$$

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{x^{2}}{36}-\frac{y^{2}}{4}=1$$

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (3,0),(3,10)\(;\) minor axis of length 4

Identify the conic and sketch its graph. $$r=\frac{3}{2+4 \sin \theta}$$

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(2,3 \pi / 4)$$

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=4$$

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=-\frac{5}{2}$$

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=4 \cos \theta\\\ &y=2 \sin \theta \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x=-1\)

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$9 x^{2}+24 x y+16 y^{2}+90 x-130 y=0$$

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{y^{2}}{1}-\frac{x^{2}}{4}=1$$