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Chapter 10

Topics in Analytic Geometry

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Problem 1

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

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Problem 2

Fill in the blanks.
If a nonvertical line has inclination $\theta$ and slope $m,$ then $m=$ ______.

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Problem 3

Fill in the blanks.
If two nonperpendicular lines have slopes $m_{1}$ and $m_{2},$ then the angle between the two lines is $\tan \theta=$ ______.

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Problem 4

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

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Problem 5

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

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

Find the slope of the line with inclination $\theta$.
$\theta=\frac{5 \pi}{6}$ radians

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Problem 11

Find the slope of the line with inclination $\theta$.
$\theta=0.26$ radian

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Problem 12

Find the slope of the line with inclination $\theta$.
$\theta=0.74$ radian

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Problem 13

Find the slope of the line with inclination $\theta$.
$\theta=1.27$ radians

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Problem 14

Find the slope of the line with inclination $\theta$.
$\theta=1.35$ radians

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Problem 15

Find the slope of the line with inclination $\theta$.
$\theta=1.81$ radians

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Problem 16

Find the slope of the line with inclination $\theta$.
$\theta=2.88$ radians

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Problem 17

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=-1$

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Problem 18

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=-2$

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Problem 19

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=1$

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Problem 20

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=2$

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Problem 21

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=\frac{3}{4}$

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Problem 22

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=\frac{1}{2}$

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Problem 23

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=-\frac{5}{2}$

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Problem 24

Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$
$m=-\frac{7}{9}$

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Problem 25

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(\sqrt{3}, 2),(0,1)$

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Problem 26

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(1,2 \sqrt{3}),(0, \sqrt{3})$

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Problem 27

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(-\sqrt{3},-1),(0,-2)$

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Problem 28

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(3, \sqrt{3}),(6,-2 \sqrt{3})$

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Problem 29

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(6,1),(10,8)$

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Problem 30

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(12,8),(-4,-3)$

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Problem 31

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(-2,20),(10,0)$

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Problem 32

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(0,100),(50,0)$

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Problem 33

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$\left(\frac{1}{4}, \frac{3}{2}\right),\left(\frac{1}{3}, \frac{1}{2}\right)$

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Problem 34

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$\left(\frac{2}{5},-\frac{3}{4}\right),\left(-\frac{11}{10},-\frac{1}{4}\right)$

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Problem 35

Find the inclination $\theta$ (in radians and degrees) of the line.
$2 x+2 y-5=0$

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Problem 36

Find the inclination $\theta$ (in radians and degrees) of the line.
$x-\sqrt{3} y+1=0$

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Problem 37

Find the inclination $\theta$ (in radians and degrees) of the line.
$3 x-3 y+1=0$

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Problem 38

Find the inclination $\theta$ (in radians and degrees) of the line.
$\sqrt{3} x-y+2=0$

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Problem 39

Find the inclination $\theta$ (in radians and degrees) of the line.
$x+\sqrt{3} y+2=0$

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Problem 40

Find the inclination $\theta$ (in radians and degrees) of the line.
$-2 \sqrt{3} x-2 y=0$

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Problem 41

Find the inclination $\theta$ (in radians and degrees) of the line.
$6 x-2 y+8=0$

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Problem 42

Find the inclination $\theta$ (in radians and degrees) of the line.
$2 x-6 y-12=0$

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Problem 43

Find the inclination $\theta$ (in radians and degrees) of the line.
$4 x+5 y-9=0$

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Problem 44

Find the inclination $\theta$ (in radians and degrees) of the line.
$5 x+3 y=0$

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Problem 45

Find the angle $\theta$ (in radians and degrees) between the lines.
$3 x+y=3$
$x-y=2$

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Problem 46

Find the angle $\theta$ (in radians and degrees) between the lines.
$x+3 y=2$
$x-2 y=-3$

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Problem 47

Find the angle $\theta$ (in radians and degrees) between the lines.
$\begin{aligned} x-y &=0 \\ 3 x-2 y &=-1 \end{aligned}$

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Problem 48

Find the angle $\theta$ (in radians and degrees) between the lines.
$2 x-y=2$
$4 x+3 y=24$

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Problem 49

Find the angle $\theta$ (in radians and degrees) between the lines.
$x-2 y=7$
$6 x+2 y=5$

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Problem 50

Find the angle $\theta$ (in radians and degrees) between the lines.
$5 x+2 y=16$
$3 x-5 y=-1$

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Problem 51

Find the angle $\theta$ (in radians and degrees) between the lines.
$x+2 y=8$
$x-2 y=2$

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Problem 52

Find the angle $\theta$ (in radians and degrees) between the lines.
$3 x-5 y=3$
$3 x+5 y=12$

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Problem 53

Find the angle $\theta$ (in radians and degrees) between the lines.
$0.05 x-0.03 y=0.21$
$0.07 x+0.02 y=0.16$

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Problem 54

Find the angle $\theta$ (in radians and degrees) between the lines.
$0.02 x-0.05 y=-0.19$
$0.03 x+0.04 y \quad =0.52$

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Problem 55

Find the slope of each side of the triangle, and use the slopes to find the measures of the interior angles.

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Problem 56

Find the slope of each side of the triangle, and use the slopes to find the measures of the interior angles.

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Problem 57

Find the slope of each side of the triangle, and use the slopes to find the measures of the interior angles.

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Problem 58

Find the slope of each side of the triangle, and use the slopes to find the measures of the interior angles.

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Problem 59

Find the distance between the point and the line.
Point $\quad$ Line
$(1,1) \quad y=x+1$

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Problem 60

Find the distance between the point and the line.
Point $\quad$ Line
$(2,1) \quad y=x+2$

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Problem 61

Find the distance between the point and the line.
Point $\quad$ Line
$(3,2) \quad y=2 x-1$

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Problem 62

Find the distance between the point and the line.
Point $\quad$ Line
$(1,4) \quad y=4 x+2$

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Problem 63

Find the distance between the point and the line.
Point $\quad$ Line
$(-2,6) \quad y=-x+5$

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Problem 64

Find the distance between the point and the line.
Point $\quad$ Line
$(4,-4) \quad y=-2 x-3$

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Problem 65

Find the distance between the point and the line.
Point $\quad$ Line
$(1,-3) \quad y=2 x-5$

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Problem 66

Find the distance between the point and the line.
Point $\quad$ Line
$(-2,8) \quad y=-3 x+2$

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Problem 67

Find the distance between the point and the line.
Point $\quad$ Line
$(2,3) \quad 3 x+y=1$

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Problem 68

Find the distance between the point and the line.
Point $\quad$ Line
$(2,1) \quad-2 x+y=2$

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Problem 69

Find the distance between the point and the line.
Point $\quad$ Line
$(6,2) \quad-3 x+4 y=-5$

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Problem 70

Find the distance between the point and the line.
Point $\quad$ Line
$(1,-3) \quad 4 x-3 y=-7$

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Problem 71

Find the distance between the point and the line.
Point $\quad$ Line
$(-1,2) \quad 5 x+3 y=-4$

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Problem 72

Find the distance between the point and the line.
Point $\quad$ Line
$(2,-3) \quad 4 x-5 y=-2$

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Problem 73

Find the distance between the point and the line.
Point $\quad$ Line
$(-1,-5) \quad 6 x+3 y=3$

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Problem 74

Find the distance between the point and the line.
Point $\quad$ Line
$(-5,-3) \quad-2 x-6 y=7$

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Problem 75

The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.
$A(-1,0), B(0,3), C(3,1)$

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Problem 76

The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.
$A(-4,0), B(0,5), C(3,3)$

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Problem 77

The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.
$A(-3,0), B(0,-2), C(2,3)$

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Problem 78

The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.
$A(-2,0), B(0,-3), C(5,1)$

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Problem 79

The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.
$A(1,1), B(2,4), C(3,5)$

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Problem 80

The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.
$A(-3,-2), B(-1,-4), C(3,-1)$

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Problem 81

Find the distance between the parallel lines.
$x+y=1$
$x+y=5$

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Problem 82

Find the distance between the parallel lines.
$3 x-4 y=1$
$3 x-4 y=10$

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Problem 83

A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

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Problem 84

A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

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Problem 85

A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof.

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Problem 86

A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel.
(a) Draw a diagram that gives a visual representation of the problem.
(b) Find the inclination of the conveyor.
(c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

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Problem 87

Find the angles $\alpha$ and $\beta$ shown in the drawing of the roof truss.

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Problem 88

The Falls Incline Railway in Niagara Falls, Ontario, Canada, is an inclined railway
that was designed to carry people from the City of Niagara Falls to Queen Victoria Park. The railway is approximately 170 feet long with a 36$\%$ uphill grade (see figure).
(a) Find the inclination $\theta$ of the railway.
(b) Find the change in elevation from the base to the top of the railway.
(c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track.
(d) Sketch a graph of the equation you found in part (c).

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Problem 89

Determine whether the statement is true or false. Justify your answer.
A line that has an inclination greater than $\pi / 2$ radians has a negative slope.

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Problem 90

Determine whether the statement is true or false. Justify your answer.
To find the angle between two lines whose angles of inclination $\theta_{1}$ and $\theta_{2}$ are known, substitute $\theta_{1}$ and $\theta_{2}$ for $m_{1}$ and $m_{2},$ respectively, in the formula for the angle between two lines.

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Problem 91

Determine whether the statement is true or false. Justify your answer.
The inclination of a line is the angle between the line and the $x$ -axis.

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Problem 92

Use the pentagon shown below.
(a) Describe how you can use the formula for the distance between a point and a line to find the area of the pentagon.
(b) Describe how you can use the formula for the angle between two lines to find the measures of the interior angles of the pentagon.

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Problem 93

Consider a line with slope $m$ and $y$ -intercept $(0,4) .$
(a) Write the distance $d$ between the origin and the line as a function of $m .$
(b) Graph the function in part (a).
(c) Find the slope that yields the maximum distance between the origin and the line.
(d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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Problem 94

Consider a line with slope $m$ and $y$ -intercept $(0,4) .$
(a) Write the distance $d$ between the point $(3,1)$ and the line as a function of $m .$
(b) Graph the function in part (a).
(c) Find the slope that yields the maximum distance between the point and the line.
(d) Is it possible for the distance to be 0$?$ If so, what is the slope of the line that yields a distance of 0$?$
(e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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Problem 95

Explain why the inclination of a line can be an angle that is greater than $\pi / 2$ , but the angle between two lines cannot be greater than $\pi / 2$ .

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