Chapter 10

Sketch the limaçon \(r=3-4 \sin \theta\), and find the area of the region inside its small loop.

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Directrix is \(y-2=0\)

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x=0\)

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(\frac{x^{2}}{7}+\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-4 y^{2}+8 x+12 y-6=0 $$

Sketch the limaçon \(r=2-4 \cos \theta\), and find the area of the region inside its small loop.

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is \(\left(0,-\frac{1}{9}\right)\)

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(y=-2\)

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=2 \sin t, y=3 \cos t ; 0 \leq t \leq 2 \pi $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(16 x^{2}+4 y^{2}=32\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-24 x+36=0 $$

Sketch the limaçon \(r=2-3 \cos \theta\), and find the area of the region inside its large loop.

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is \((-4,0)\)

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x-y=0\)

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=3 \sin r, y=-2 \cos r ; 0 \leq r \leq 2 \pi $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(4 x^{2}+25 y^{2}=100\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-24 x+35=0 $$

Sketch one leaf of the four-leaved rose \(r=3 \cos 2 \theta\), and find the area of the region enclosed by it.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=4-3 \cos \theta\) (limaçon)

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Directrix is \(y=\frac{7}{2}\)

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x^{2}+y^{2}=4\)

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(10 x^{2}-25 y^{2}=100\)

Sketch the graph of the given equation. $$ \frac{(x+3)^{2}}{4}+\frac{(y+2)^{2}}{16}=1 $$

Sketch the three-leaved rose \(r=4 \cos 3 \theta\), and find the area of the total region enclosed by it.

Find the equation of the parabola with vertex at the origin and axis along the \(x\)-axis if the parabola passes through the point \((3,-1)\). Make a sketch.

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x^{2}=4 p y\)

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(x^{2}-4 y^{2}=8\)

Sketch the graph of the given equation. $$ (x+3)^{2}+(y-4)^{2}=25 $$

Sketch the three-leaved rose \(r=2 \sin 3 \theta\), and find the area of the region bounded by it.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=5-3 \cos \theta\) (limaçon)

Find the equation of the parabola through the point \((-2,4)\) if its vertex is at the origin and its axis is along the \(x\)-axis. Make a sketch.

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=9 \sin ^{2} \theta, y=9 \cos ^{2} \theta ; 0 \leq \theta \leq \pi $$

Find the equation of the given central conic. Ellipse with a focus at \((-3,0)\) and a vertex at \((6,0)\)

Sketch the graph of the given equation. $$ \frac{(x+3)^{2}}{4}-\frac{(y+2)^{2}}{16}=1 $$

Find the area of the region between the two concentric circles \(r=7\) and \(r=10\).

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r^{2}=4 \cos 2 \theta\) (lemniscate)

Find the equation of the parabola through the point \((6,-5)\) if its vertex is at the origin and its axis is along the \(y\)-axis. Make a sketch.

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r=3\)

Sketch the graph of the given equation. $$ 4(x+3)=(y+2)^{2} $$

Sketch the region that is inside the circle \(r=3 \sin \theta\) and outside the cardioid \(r=1+\sin \theta\), and find its area.

Find the equation of the parabola whose vertex is the origin and whose axis is the \(y\)-axis if the parabola passes through the point \((-3,5)\). Make a sketch.

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r \cos \theta+3=0\)

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=\cos \theta, y=-2 \sin ^{2} 2 \theta ;-\infty<\theta<\infty $$

Find the equation of the given central conic. Ellipse with a focus at \((0,-5)\) and eccentricity \(\frac{1}{3}\)

Sketch the graph of the given equation. $$ (x+2)^{2}=8(y-1) $$

Sketch the region that is outside the circle \(r=2\) and inside the lemniscate \(r^{2}=8 \cos 2 \theta\), and find its area.

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$y^{2}=16 x,(1,-4)$$

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r-5 \cos \theta=0\)

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=\sin \theta, y=2 \cos ^{2} 2 \theta ;-\infty<\theta<\infty $$

Sketch the graph of the given equation. $$ (x+2)^{2}=4 $$