Precalculus

John W. Coburn

Chapter 9

Analytical Geometry and the Conic Sections - all with Video Answers

Educators

Section 1

A Brief Introduction to Analytical Geometry

Problem 1

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
Analytical geometry is a study of ______ using the tools of _______ .

Jennifer Durham

Jennifer Durham

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Problem 2

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
The distance formula is $d=$ ______ the midpoint formula is $M=$ _____

Jennifer Durham

Jennifer Durham

Numerade Educator

Problem 3

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
The distance between a point and a line always refers to the _____distance.

Jennifer Durham

Jennifer Durham

Numerade Educator

Problem 4

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
The conic sections are formed by the intersection of a _____ and a ______.

Jennifer Durham

Jennifer Durham

Numerade Educator

Problem 5

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
If a plane intersects a cone at its vertex, the result is a _________ , a line, or a pair of _______ lines.

Jennifer Durham

Jennifer Durham

Numerade Educator

Problem 6

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
A circle is defined relative to an equal distance between two ______ . A parabola is defined relative to an equal distance between a ______ and a _______.

Jennifer Durham

Jennifer Durham

Numerade Educator

Problem 7

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(-5,2)\\&P_{2}=(1,2)\\&P_{3}=(-5,-6)\end{aligned}$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 8

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(3,2)\\&P_{2}=(3,14)\\&P_{3}=(8,2)\end{aligned}$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 9

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(-2,1)\\&P_{2}=(6,-5)\\&P_{3}=(2,-7)\end{aligned}$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 10

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(0,-5)\\&P_{2}=(-6,4)\\&P_{3}=(6,-1)\end{aligned}$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 11

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(10,-21)\\&P_{2}=(-6,-9)\\&P_{3}=(3,3)\end{aligned}$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 12

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(6,-6)\\&P_{2}=(-12,18)\\&P_{3}=(20,42)\end{aligned}$$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 13

Find the equation of the circle that circumscribes the triangle in Exercise 7

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 14

Find the equation of the circle that circumscribes the triangle in Exercise 8

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 15

Find the equation of the circle that circumscribes the triangle in Exercise 9

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 16

Find the equation of the circle that circumscribes the triangle in Exercise 10

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 17

Find the equation of the circle that circumscribes the triangle in Exercise 11

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 18

Find the equation of the circle that circumscribes the triangle in Exercise 12

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 19

Of the following six points, four are an equal distance from the point $A(2,3)$ and two are not.
(a) Identify which four, and (b) find any two additional points that are this same (nonvertical, nonhorizontal) distance from ( 2,3 ): $B(7,15) \quad C(-10,8) \quad D(9,14) \quad E(-3,-9)$ $F(5,4+3 \sqrt{10}) \quad G(2-2 \sqrt{30}, 10)$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 20

Of the following six points, four are an equal distance from the point $P(-1,4)$ and two are not.
(a) Identify which four, and (b) Find any two additional points that are the same (nonvertical, nonhorizontal) distance from (-1,4) $Q(-9,10) \quad R(5,12) \quad S(-7,11) \quad T(4,4+5 \sqrt{3})$ $U(-1+4 \sqrt{6}, 6) \quad V(-7,4+\sqrt{51})$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 21

The Perpendicular Distance from a Point to a Line: $d=\left|\frac{A x_{1}+B y_{1}+C}{\sqrt{A^{2}+B^{2}}}\right|$ The perpendicular distance from a point $\left(x_{1}, y_{1}\right)$ to a given line can be found using the formula shown, where $A x+B y+C=0$ is the equation of the line in standard form $(A, B, \text { and } C$ are integers).
Use the formula to verify that $P(-6,2)$ and $Q(6,4)$ are an equal distance from the line $y=-\frac{1}{2} x+3$ .

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 22

The Perpendicular Distance from a Point to a Line: $d=\left|\frac{A x_{1}+B y_{1}+C}{\sqrt{A^{2}+B^{2}}}\right|$ The perpendicular distance from a point $\left(x_{1}, y_{1}\right)$ to a given line can be found using the formula shown, where $A x+B y+C=0$ is the equation of the line in standard form $(A, B, \text { and } C$ are integers).
Find the value(s) for $y$ that ensure $(1, y)$ is this same distance from $y=-\frac{1}{2} x+3$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 23

Of the following four points, three are an equal distance from the point $A(0,1)$ and the line $y=-1 .$ (a) Identify which three, and (b) find any two additional points that satisfy these conditions. $B(-6,9) \quad C(4,4) \quad D(-2 \sqrt{2}, 6) \quad E(4 \sqrt{2}, 8)$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 24

Of the following four points, three are an equal distance from the point $P(2,4)$ and the line $y=-4 .$ (a) Identify which three, and (b) find any two additional points that satisfy these conditions. $Q(-10,9) \quad R(2+4 \sqrt{2}, 3) \quad S(10,4)$
$T(2-4 \sqrt{5}, 5)$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 25

Consider a fixed point (0,-4) and a fixed line $y=4 .$ Verify that the distance from each point to $(0,-4),$ is equal to the distance from the point to the line $y=4$ $A(4,-1) \quad B\left(10,-\frac{25}{4}\right) \quad C(4 \sqrt{2},-2)$ $D(8 \sqrt{5},-20)$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 26

Consider a fixed point (0,-2) and a fixed line $y=2 .$ Verify that the distance from each point to $(0,-2),$ is equal to the distance from the point to the line $y=2$ $P(12,-18) \quad Q\left(6,-\frac{9}{2}\right) \quad R(4 \sqrt{5},-10)$ $S(4 \sqrt{6},-12)$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 27

The points from Exercise 25 are on the graph of a parabola. Find the equation of the parabola.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 28

The points from Exercise 26 are on the graph of a parabola. Find the equation of the parabola.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 29

Using (0,-2) as the focus and the horizontal line $y=-8$ as the directrix, find an equation for the set of all points $(x, y)$ where the distance from the focus to $(x, y)$ is one-half the distance from the directrix to $(x, y)$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 30

Using (4,0) as the focus and the vertical line $x=9$ as the directrix, find an equation for the set of all points $(x, y)$ where the distance from the focus to
$(x, y)$ is two-thirds the distance from the directrix to $(x, y)$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 31

From Exercise $29,$ verify the points (-3,2) and $(\sqrt{12}, 0)$ are on the ellipse defined by $4 x^{2}+3 y^{2}=48 .$ Then verify that $d_{1}+d_{2}=d_{3}+d_{4}$ . (GRAPH CAN'T COPY)

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 32

From Exercise $30,$ verify the points $\left(4, \frac{10}{3}\right)$ and $(-3,-\sqrt{15})$ are on the ellipse defined by $5 x^{2}+9 y^{2}=180$ Then verify that $d_{1}+d_{2}=d_{3}+d_{4}$. (GRAPH CAN'T COPY)

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 33

From the focus/directrix definition of a hyperbola: If the distance from the focus to a point $(x, y)$ is greater than the distance from the directrix to $(x, y)$
one branch of a hyperbola is formed. Using (2,0) as the focus and the vertical line $x=\frac{1}{2}$ as the directrix, find an equation for the set of all points $(x, y)$ where the distance from the focus to $(x, y),$ is twice the distance from the directrix to $(x, y)$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 34

From the two foci definition of a hyperbola: For foci $f_{1}$ and $f_{2},$ a hyperbola is the set of all points $(x, y)$ where the difference of the distances from $f_{1}$ to $(x, y)$ and $f_{2}$ to $(x, y)$ is constant. Verify the points (2,3) and $(-3,-2 \sqrt{6})$ are on the graph of the hyperbola from Exercise $33 .$ Then verify $d_{1}-d_{2}=d_{3}-d_{4}$ (GRAPH CAN'T COPY)

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 35

Do some reading or research on the orthocenter of a triangle, and the centroid of a triangle. How are they found? What are their properties? Use the ideas and skills from this section to find the (a) orthocenter and (b) centroid of the triangle formed by the points $A(-8,2), B(-2,-6),$ and $C(4,0).$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 36

Properties of a circle: $A$ theorem from elementary geometry states: If a radius is perpendicular to a chord, it bisects the chord. Verify this is true for the circle, radii, and chords shown. (GRAPH CAN'T COPY)

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 37

Verify that points $C(-2,3)$ and $D(2 \sqrt{2}, \sqrt{6})$ are points on the ellipse with foci at $A(-2,0)$ and $B(2,0),$ by verifying $d(A C)+d(B C)=d(A D)+d(B D) .$ The expression that results has the form $\sqrt{A+B}+\sqrt{A-B},$ which prior to the common use of technology had to be simplified using the formula $\sqrt{A+B}+\sqrt{A-B}=\sqrt{a+\sqrt{b}}$ where $a=2 A$ and $b=4\left(A^{2}-B^{2}\right) .$ Use this relationship to verify the equation above.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 38

Verify the following is an identity: $\frac{\cos (2 x)+\sin ^{2} x}{1-\cos ^{2} x}=\cot ^{2}(x)$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 39

Find all solutions in $[0,2 \pi)$ $-225=600+825 \sin \left(x+\frac{\pi}{6}\right)$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 40

Solve for $x$ in both exact and approximate form:
a. $5=\frac{10}{1+9 e^{-0.5 x}}$
b. $345=5 e^{0.4 x}+75$

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 41

Sketch a complete graph of $h(x)=\frac{x^{2}-9}{x^{2}-4}$ Clearly label all intercepts and asymptotes.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator