Chapter 10

Fill in the blank: The locus of a point in the plane that moves such that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a _____.

Fill in the blank. The equation \(r=2+\cos \theta\) represents a _____.

Fill in the blank(s). The origin of the polar coordinate system is called the _____.

fill in the blank(s). A _______ is the intersection of a plane and a double-napped cone.

A _____ is the set of all points \((x, y)\) in a plane for which the absolute value of the difference of the distances from two distinct fixed points, called foci, is constant.

An __________ is the set of all points \((x, y)\) in a plane, the sum of whose distances from two distinct fixed points called _________ is constant.

Match the conic with its eccentricity. (a) \(01\) (i) ellipse (ii) hyperbola (iii) parabola

Fill in the blank. The equation \(r=2 \cos \theta\) represents a ______.

Fill in the blank(s). The _____ of a curve is the direction in which the curve is traced out for increasing values of the parameter.

Fill in the blank(s). For the point \((r, \theta), r\) is the ______ from \(O\) to \(P\) and \(\theta\) is the ______ counterclockwise from the polar axis to segment \(\overline{O P}\).

fill in the blank(s). A collection of points satisfying a geometric property can also be referred to as a _______ of points.

The line segment connecting the vertices of a hyperbola is called the ______ and the midpoint of the line segment is the ______ of the hyperbola.

The chord joining the vertices of an ellipse is called the _________ ,and its midpoint is the __________ of the ellipse.

A conic has a polar equation of the form \(r=\frac{e p}{1+e \cos \theta} .\) Is the directrix vertical or horizontal?

Fill in the blank. The equation \(r^{2}=4 \sin 2 \theta\) represents a ______.

Given a set of parametric equations, how do you find the corresponding rectangular equation?

How are the rectangular coordinates \((x, y)\) related to the polar coordinates \((r, \theta) ?\)

The form \(\frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1\) represents a hyperbola with center at what point?

The chord perpendicular to the major axis at the center of an ellipse is called the __________ of the ellipse.

A conic with a horizontal directrix has a polar equation of the form \(r=\frac{e p}{1-e \sin \theta} \cdot\) Is the directrix above or below the pole?

Fill in the blank. The equation \(r=1+\sin \theta\) represents a ______.

What point on the plane curve represented by the parametric equations \(x=t\) and \(y=t\) corresponds to \(t=3 ?\)

Do the polar coordinates \((1, \pi)\) and the rectangular coordinates (-1,0) represent the same point?

fill in the blank(s). A _______ is the set of all points $$(x, y)$$ in a plane that are equidistant from a fixed line, called the _______ , and a fixed point, called the _______ , not on the line.

How many asymptotoes does a hyperbola have?

Use a graphing utility to graph the polar equation for (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic for each equation. $$r=\frac{2 e}{1+e \cos \theta}$$

How can you test whether the graph of a polar equation is symmetric with respect to the line \(\theta=\frac{\pi}{2} ?\)

What does the equation \((x-h)^{2}+(y-k)^{2}=r^{2}\) represent? What do \(h, k,\) and \(r\) represent?

Where do the asymptotes of a hyperbola intersect?

Consider the ellipse given by \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1.\) Is the major axis horizontal or vertical?

Use a graphing utility to graph the polar equation for (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic for each equation. $$r=\frac{2 e}{1-e \cos \theta}$$

Is the graph of \(r=3+4 \cos \theta\) symmetric with respect to the line \(\theta=\frac{\pi}{2}\) or to the polar axis?

The tangent line to a parabola at a point \(P\) makes equal angles with what two lines?

What type of conic does \(A x^{2}+C y^{2}+D x+E y+F=0\) represent when \(A C>0 ?\)

Consider the ellipse given by \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1.\) What is the length of the major axis?

Use a graphing utility to graph the polar equation for (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic for each equation. $$r=\frac{2 e}{1-e \sin \theta}$$

Find the standard form of the equation of the circle with the given characteristics. Center at origin; radius: 4

Consider the ellipse given by \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1.\) What is the length of the minor axis?

Use a graphing utility to graph the polar equation for (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic for each equation. $$r=\frac{2 e}{1+e \sin \theta}$$

Find the standard form of the equation of the circle with the given characteristics. Center at origin; radius: \(\sqrt{11}\)

Consider the ellipse given by \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1.\) Is the ellipse elongated or nearly circular?

Consider the parametric equations \(x=\sqrt{t}\) and \(y=2-t.\) (a) Create a table of \(x\) - and \(y\) -values using \(t\)=0,1, 2, 3, and 4. (b) Plot the points \((x, y)\) generated in part (a) and sketch a graph of the parametric equations for \(0 \leq t \leq 4.\) Describe the orientation of the curve. (c) Use a graphing utility to graph the curve represented by the parametric equations. (d) Find the rectangular equation by eliminating the parameter. Sketch its graph. How does the graph differ from those in parts (b) and (c)?

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(3, \frac{5 \pi}{6}\right)$$

Find the standard form of the equation of the circle with the given characteristics. Center: (3,7)\(;\) point on circle: (1,0)

Consider the parametric equations \(x=4 \cos ^{2} t\) and \(y=4 \sin t.\) (a) Create a table of \(x\)- and \(y\)-values using \(t=-\pi / 2\) \(-\pi / 4,0, \pi / 4,\) and \(\pi / 2.\) (b) Plot the points \((x, y)\) generated in part (a) and sketch a graph of the parametric equations for \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}.\) Describe the orientation of the curve. (c) Use a graphing utility to graph the curve represented by the parametric equations. (d) Find the rectangular equation by eliminating the parameter. (Hint: Use the trigonometric identity \(\left.\cos ^{2} t+\sin ^{2} t=1 .\right)\) Sketch its graph. How does the graph differ from those in parts (b) and (c)?

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(2, \frac{3 \pi}{4}\right)$$

Find the standard form of the equation of the circle with the given characteristics. Center: (6,-3)\(;\) point on circle: (-2,4)

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(1,-\frac{\pi}{3}\right)$$

Find the standard form of the equation of the circle with the given characteristics. Endpoints of a diameter: (-6,0) and (0,-2)

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,±2)\(;\) foci: (0,±4)