Chapter 10

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 blanks. The origin of the polar coordinate system is called the ____.

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

Fill in the blanks. If \(f\) and \(g\) are continuous functions of \(t\) on an interval \(I\), then the set of ordered pairs \((f(t), g(t))\) is a _____ _____ \(C\).

Fill in the blanks. 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 ________, 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.

Fill in the blanks. 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 \(\overline{O P}\).

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

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

Fill in the blanks. The graph of a hyperbola has two disconnected parts called ________.

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

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

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

Fill in the blanks. To plot the point \((r, \theta),\) use the _____ coordinate system.

Fill in the blanks. The process of converting a set of parametric equations to a corresponding rectangular equation is called _____ the ______.

Fill in the blanks. The line segment connecting the vertices of a hyperbola is called the ________ ________, and the midpoint of the line segment is the ________ of the hyperbola.

Fill in the blanks. A collection of points satisfying a geometric property can also be referred to as ______of points.

The chord perpendicular to the major axis at the center of the ellipse is called the _______________ _____________ of the ellipse.

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

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

Fill in the blanks. The polar coordinates \((r, \theta)\) are related to the rectangular coordinates \((x, y)\) as follows: \(x=\) ____ \(y=\) ____ \(\tan \theta=\) _____ \(r^{2}=\) ____

The distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+C=0\) is given by \(d=\)_____.

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

Fill in the blanks. The quantity \(B^{2}-4 A C\) is called the _________ of the equation \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\).

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

Fill in the blanks. 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 concept of ____________ is used to measure the ovalness of an ellipse.

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. $$r=\frac{2 e}{1+e \cos \theta}$$

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(2,5 \pi / 6)$$

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

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{\pi}{6} \text { radian }$$

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?

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

Fill in the blanks. The line that passes through the focus and the vertex of a parabola is called the _______ of the parabola.

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. $$r=\frac{2 e}{1-e \cos \theta}$$

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(3,5 \pi / 4)$$

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

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{\pi}{4} \text { radian }$$

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?

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

Fill in the blanks. The _______ of a parabola is the midpoint between the focus and the directrix.

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. $$r=\frac{2 e}{1-e \sin \theta}$$

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(4,-\pi / 3)$$

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{3 \pi}{4} \text { radians }$$

(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. $$\begin{aligned} &x=t\\\ &y=-4 t \end{aligned}$$

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

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

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. $$r=\frac{2 e}{1+e \sin \theta}$$

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(-1,-3 \pi / 4)$$

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{2 \pi}{3} \text { radians }$$