Chapter 10

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is at \((2,0)\)

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r^{2}=a \cos 2 \theta, a>0 $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}-4 y^{2}-2 x+2 y+1=0

Find polar coordinates of the points whose Cartesian coordinates are given. (a) \((-3 / \sqrt{3}, 1 / \sqrt{3})\) (b) \((-\sqrt{3} / 2, \sqrt{3} / 2)\) (c) \((0,-2)\) (d) \((3,-4)\)

a parametric representation of a curve is given. $$ x=t^{3}-2 t, y=t^{2}-2 t ;-3 \leq t \leq 3 $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ \frac{x^{2}}{16}-\frac{y^{2}}{4}=1 $$

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Directrix is \(x=3\)

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

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}-4 y^{2}+8 x+12 y-5=0

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x-3 y+2=0 $$

a parametric representation of a curve is given. $$ x=2 \sqrt{t-2}, y=3 \sqrt{4-t} ; 2 \leq t \leq 4 $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ \frac{-x^{2}}{9}+\frac{y^{2}}{4}=1 $$

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

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

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}-4 y^{2}+8 x+12 y-6=0

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x=0 $$

a parametric representation of a curve is given. $$ x=3 \sqrt{t-3}, y=2 \sqrt{4-t} ; 3 \leq t \leq 4 $$

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

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

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

Sketch the graph of the given polar equation and verify its symmetry (see Examples \(1-3)\). \(r=1-2 \sin \theta\) (limaçon)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}-24 x+36=0

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ y=-2 $$

a parametric representation of a curve is given. $$ x=2 \sin t, y=3 \cos t ; 0 \leq t \leq 2 \pi $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ 16 x^{2}+4 y^{2}=32 $$

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

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

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

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}-24 x+35=0

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x-y=0 $$

a parametric representation of a curve is given. $$ x=3 \sin r, y=-2 \cos r ; 0 \leq r \leq 2 \pi $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ 4 x^{2}+25 y^{2}=100 $$

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

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.

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x^{2}+y^{2}=4 $$

a parametric representation of a curve is given. $$ x=-2 \sin r, y=-3 \cos r ; 0 \leq r \leq 4 \pi $$

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ 10 x^{2}-25 y^{2}=100 $$

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.

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.

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 (if it is a hyperbola). $$ x^{2}-4 y^{2}=8 $$

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.

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 \(r=10\)

Find the Cartesian equations of the graphs of the given polar equations. $$ \theta=\frac{1}{2} \pi $$

a parametric representation of a curve is given. $$ x=9 \sin ^{2} \theta, y=9 \cos ^{2} \theta ; 0 \leq \theta \leq \pi $$

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

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.