Chapter 10

The locus of a point in the plane that moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a ________.

The graph of \(r=f(\sin\ \theta)\) is symmetric with respect to the line ________.

The origin of the polar coordinate system is called the ________.

The procedure used to eliminate the \(xy\)-term in a general second-degree equation is called ________ of ________.

A ________ is the set of all points \((x, y)\) in a plane, the difference of whose distances from two distinct fixed points, called ________, is a positive constant.

A ________ is the intersection of a plane and a double-napped cone.

The ________ of a nonhorizontal line is the positive angle \(\theta\) (less than \(\pi\)) measured counterclockwise from the \(x\)-axis to the line.

For the point \((r, \theta), r\) is the ________ ________ from \(O\) to \(P\) and \(\theta\) is the ________ ________ , counterclockwise from the polar axis to the line segment \(\overline{OP}\).

The ________ of a curve is the direction in which the curve is traced out for increasing values of the parameter.

After rotating the coordinate axes through an angle \(\theta\), the general second-degree equation in the new \(x'y'\)-plane will have the form ________.

The graph of a hyperbola has two disconnected parts called ________.

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

When a plane passes through the vertex of a double-napped cone, the intersection is a ________ ________.

If a nonvertical line has inclination \(\theta\) and slope \(m\), then \(m =\) ______________ .

An equation of the form \(r=\dfrac{ep}{1+e\cos\ \theta}\) has a ________ directrix to the ________ of the pole.

The equation \(r=2\ + \cos\ \theta\) represents a ________ ________.

To plot the point \((r, \theta),\) use the ________ coordinate system.

The process of converting a set of parametric equations to a corresponding rectangular equation is called ________ the ________.

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 perpendicular to the major axis at the center of the ellipse is called the ________ ________ of the ellipse.

A collection of points satisfying a geometric property can also be referred to as a ________ of points.

Match the conic with its eccentricity. (a) \(e<1 \quad \quad \quad \quad \quad \) (b) \(e=1 \quad \quad \quad \quad \quad \) (c) \(e>1\) (i) \(\textrm{parabola} \quad \quad \quad \quad\) (ii) \(\textrm{hyperbola} \quad \quad \quad \quad\) (iii) \(\textrm{ellipse}\)

The equation \(r=2\ \cos\ \theta\) represents a ________.

The polar coordinates \((r, 0)\) are related to the rectangular coordinates \((x, y)\) as follows: \(x =\) ________ \(\quad y =\) ________ \(\quad \tan\ \theta =\) ________ \(\quad r^2 =\) ________

A curve traced by a point on the circumference of a circle as the circle rolls along a straight line in a plane is called a ________.

The quantity \(B\)^2-4AC\( is called the ________ of the equation \)Ax^2+Bxy+Cy^2+Dx+Ey+F=0$.

Each hyperbola has two ________ that intersect at the center of the hyperbola.

The concept of ________ is used to measure the ovalness of an ellipse.

A ________ is defined as 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.

The distance between the point \((x_1, y_1)\) and the line \(Ax +By + C = 0\) is given by \(d =\) ________ .

In Exercises 5-8, write the polar equation of the conic for \(e = 1. e = 0.5,\) and \(e = 1.5.\) Identify the conic for each equation. Verify your answers with a graphing utility. \(r=\dfrac{2e}{1+e\ \cos\ \theta}\)

The equation \(r^2 =4\ \sin\ 2\theta\) represents a ________.

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(2, \dfrac{5\pi}{6}\right)\)

Consider the parametric equations \(x=\sqrt{t}\) and \(y=3-t\). (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. (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ?

In Exercises 5-12, the \(x'y'\)-coordinate system has been rotated \(\theta\) degrees from the \(xy\)-coordinate system. The coordinates of a point in the \(xy\)-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. \(\theta=90^{\circ}, (0, 3)\)

The line that passes through the focus and the vertex of a parabola is called the ________ of the parabola.

The equation \(r =1\ +\ \sin\ \theta\) represents a ________.

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(3, \dfrac{5\pi}{4}\right)\)

Consider the parametric equations \(x=4\ \cos^2\ \theta\) and \(y=2\ \sin\ \theta\). (a) Create a table of \(x\)- and \(y\)-values using \(\theta = -\pi/2, -\pi/4, 0, \pi/4,\) and \(\pi/2\). (b) Plot the points \((x, y)\) generated in part (a), and sketch a graph of the parametric equations. (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ?

In Exercises 5-12, the \(x'y'\)-coordinate system has been rotated \(\theta\) degrees from the \(xy\)-coordinate system. The coordinates of a point in the \(xy\)-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. \(\theta=90^{\circ}, (2, 2)\)

The ________ of a parabola is the midpoint between the focus and the directrix.

In Exercises 7-12, identify the type of polar graph. \(r=5\ \cos\ 2\theta\)

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(4, -\dfrac{\pi}{3}\right)\)

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x=t-1\) \(y=3t+1\)

In Exercises 5-12, the \(x'y'\)-coordinate system has been rotated \(\theta\) degrees from the \(xy\)-coordinate system. The coordinates of a point in the \(xy\)-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. \(\theta=30^{\circ}, (1, 3)\)

A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a ________ ________ .

In Exercises 7-12, identify the type of polar graph. \(r=5 - 5\ \sin\ \theta\)

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(-1, -\dfrac{3\pi}{4}\right)\)

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x=3-2t\) \(y=2+3t\)

In Exercises 5-12, the \(x'y'\)-coordinate system has been rotated \(\theta\) degrees from the \(xy\)-coordinate system. The coordinates of a point in the \(xy\)-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. \(\theta=30^{\circ}, (2, 4)\)