chapter 9 student notes

Chapter 9 

Section 9.1 

Topics in Analytic Geometry 

Circles and Parabolas 

Course Number 

Instructor 

Objective: In this lesson you learned how to recognize conics, write 

equations of circles in standard form, write equations of 

parabolas in standard form, and use the reflective property of 

parabolas to solve problems. 

Date 

Important Vocabulary 

Define each term or concept. 

Directrix A fixed line in the plane from which each point on a parabola is the same 

distance as the distance from the point to a fixed point in the plane. 

Focus A fixed point in the plane from which each point on a parabola is the same 

distance as the distance from the point to a fixed line in the plane. 

Tangent A line is tangent to a parabola at a point on the parabola if the line intersects, 

but does not cross, the parabola at the point. 

I. Conics (Page 660) 

A conic section, or conic, is . . . 

and a double-napped cone. 

Name the four basic conic sections: 

and hyperbola. 

the intersection of a plane 

circle, ellipse, parabola, 

What you should learn 

How to recognize a conic 

as the intersection of a 

plane and a doublenapped 

cone 

In the formation of the four basic conics, the intersecting plane 

does not pass through the vertex of the cone. When the plane 

does pass through the vertex, the resulting figure is a(n) 

degenerate conic , such as 

a point, a line, or a pair of intersecting lines 

. 

II. Circles (Pages 661−662) 

A circle is the set of all points (x, y) in a plane that are 

equidistant from a fixed point (h, k), called the 

center of the circle. The distance r between the 

center and any point (x, y) on the circle is the radius . 


How to write equations 

of circles in standard 

form 

Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE 

Copyright © Houghton Mifflin Company. All rights reserved. 149

150 Chapter 9 Topics in Analytic Geometry 

The standard form of the equation of a circle with center 

(h, k) and radius r is (x − h) 2 + (y − k) 2 = r 2 . 

The standard form of the equation of a circle with radius r and 

whose center is the origin is x 2 + y 2 = r 2 . 

Example 1: The point (0, 1) is on a circle whose center is 

(−2, 1), as shown in the figure. Write the standard 

form of the equation of the circle. 

(x + 2) 2 + (y − 1) 2 = 4 

5 

y 

3 

1 

-5 -3 -1 

-1 

1 3 x 

5 

-3 

-5 

III. Parabolas (Pages 663−665) 

A parabola is . . . the set of all points (x, y) in a plane that 

are equidistant from a fixed line, the directrix, and a fixed point, 

the focus, not on the line. 



of parabolas in standard 

form 

The midpoint between the focus and the directrix is the 

vertex of a parabola. The line passing through the 

focus and the vertex is the axis of the parabola. 

The standard form of the equation of a parabola with a vertical 

axis having a vertex at (h, k) and directrix y = k − p is 

(x − h) 2 = 4p(y − k), p ≠ 0 

The standard form of the equation of a parabola with a horizontal 

axis having a vertex at (h, k) and directrix x = h − p is 

(y − k) 2 = 4p(x − h), p ≠ 0 

The focus lies on the axis p units (directed distance) from the 

vertex. If the vertex is at the origin (0, 0), the equation takes one 

of the following forms: 

x 2 = 4py, vertical axis or y 2 = 4px, horizontal axis 


Copyright © Houghton Mifflin Company. All rights reserved.

Section 9.1 Circles and Parabolas 151 

Example 2: Find the standard form of the equation of the 

parabola with vertex at the origin and focus (1, 0). 

y 2 = 4x 

IV. Reflective Property of Parabolas (Pages 665−666) 

Describe a real-life situation in which parabolas are used. 

Answers will vary. 


How to use the reflective 

property of parabolas to 

solve real-life problems 

A focal chord is . . . a line segment that passes through the 

focus of a parabola and has endpoints on the parabola. 

The specific focal chord perpendicular to the axis of a parabola 

is called the latus rectum . 

The reflective property of a parabola states that the tangent line 

to a parabola at a point P makes equal angles with the following 

two lines: 

1) The line passing through P and the focus 

2) The axis of the parabola 

y 

y 

y 

x 

x 

x 




Additional notes 

y 

y 

y 

x 

x 

x 

y 

y 

y 

x 

x 

x 

Homework Assignment 

Page(s) 

Exercises 



Section 9.2 Ellipses 153 

Section 9.2 

Ellipses 

Objective: In this lesson you learned how to write the standard form of 

the equation of an ellipse, and analyze and sketch the graphs 

of ellipses. 

Course Number 

Instructor 

Date 



Foci Distinct fixed points in the plane such that the sum of the distances from each 

point on an ellipse is constant. 

Vertices Points of intersection of an ellipse and the line through its foci. 

Major axis The chord connecting the vertices of an ellipse. 

Center The midpoint of the major axis of an ellipse. 

Minor axis The chord perpendicular to the major axis at the center of an ellipse. 

I. Introduction (Pages 671−674) 

An ellipse is . . . the set of all points (x, y) in a plane, the 

sum of whose distances from two distinct fixed points (foci) is 

constant. 



of ellipses in standard 

form 

The standard form of the equation of an ellipse with center (h, k) 

and a horizontal major axis of length 2a and a minor axis of 

length 2b, where 0 

The standard form of the equation of an ellipse with center (h, k) 

and a vertical major axis of length 2a and a minor axis of length 

2b, where 0 

In both cases, the foci lie on the major axis, c units from the 

center, with c 2 = a 2 − b 2 . 

If the center is at the origin (0, 0), the equation takes one of the 

following forms: x 2 /a 2 + y 2 /b 2 = 1 or 

x 2 /b 2 + y 2 /a 2 = 1 . 




Example 1: Sketch the ellipse given by 4x + 25y 

= 100 . 

5 

y 

2 

2 

3 

1 

x 

-5 -3 -1 1 3 5 

-1 

-3 

-5 

II. Applications of Ellipses (Page 675) 

Describe a real-life application in which parabolas are used. 



How to use properties of 

ellipses to model and 

solve real-life problems 

III. Eccentricity (Page 676) 

Eccentricity measures the ovalness of an ellipse. It is 

given by the ratio e = c/a . For every ellipse, the 

value of e lies between 0 and 1 . For an 

elongated ellipse, the value of e is close to 1 . 


How to find eccentricities 

of ellipses 

y 

y 

y 

x 

x 

x 


Page(s) 

Exercises 



Section 9.3 Hyperbolas 155 

Section 9.3 

Hyperbolas 

Objective: In this lesson you learned how to write the standard form of 

the equation of a hyperbola, and analyze and sketch the 

graphs of hyperbolas. 

Course Number 

Instructor 

Date 



Branches The two disconnected parts of the graph of a hyperbola. 

Transverse axis The line segment connecting the vertices of a hyperbola. 

Conjugate axis The line segment in a hyperbola of length 2b joining (h, k + b) and 

(h, k − b) [or (h − b, k) and (h + b, k)]. 


A hyperbola is . . . the set of all points (x, y) in a plane the 

difference of whose distances from two distinct points (foci) is a 

positive constant. 



of hyperbolas in standard 

form 

The line through a hyperbola’s two foci intersects the hyperbola 

at two points called vertices . 

The midpoint of a hyperbola’s transverse axis is the 

center of the hyperbola. 

The standard form of the equation of a hyperbola centered at 

(h, k) and having a horizontal transverse axis is 

(x − h) 2 /a 2 − (y − k) 2 /b 2 = 1 

The standard form of the equation of a hyperbola centered at 

(h, k) and having a vertical transverse axis is 

(y − k) 2 /a 2 − (x − h) 2 /b 2 = 1 

In each case, the vertices and foci are, respectively, a and c units 

from the center. Moreover, a, b, and c are related by the equation 

c 2 = a 2 + b 2 . 

If the center of the hyperbola is at the origin (0, 0), the equation 

takes one of the following forms: x 2 /a 2 − y 2 /b 2 = 1 or 

y 2 /a 2 − x 2 /b 2 = 1 . 




II. Asymptotes of a Hyperbola (Pages 682−684) 

The asymptotes of a hyperbola with a horizontal transverse axis 

are y = k ± b/a (x − h) . 

The asymptotes of a hyperbola with a vertical transverse axis 

are y = k ± a/b (x − h) . 


How to find asymptotes 

of and graph hyperbolas 

y 

Example 1: Sketch the graph of the hyperbola given by 

2 9 2 = 

y − x 9 . 

x 

The eccentricity of a hyperbola is e = c/a , where 

the values of e are greater than 1 . 

III. Applications of Hyperbolas (Page 685) 

Describe a real-life application in which hyperbolas occur or are 

used. 



How to use properties of 

hyperbolas to solve reallife 

problems 

IV. General Equations of Conics (Page 686) 

2 2 

The graph of Ax + Bxy + Cy + Dx + Ey + F = 0 is one of the 

following: 

1) Circle if A = C 

2) Parabola if AC = 0 

3) Ellipse if AC > 0 

4) Hyperbola if AC < 0 


How to classify conics 

from their general 

equations 

Example 2: Classify the equation 9x 

+ y −18x 

− 4y 

+ 4 = 0 

as a circle, a parabola, an ellipse, or a hyperbola. 

Ellipse 

2 

2 


Page(s) 

Exercises 



Section 9.4 Rotation and Systems of Quadratic Equations 157 

Section 9.4 

Rotation and Systems of Quadratic Equations 

Objective: In this lesson you learned how to eliminate the xy-term in 

equations of conics and classify conics. 

Course Number 

Instructor 

Date 



Discriminant The quantity B 2 − 4AC, of the general conic equation 

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, which can be used to classify the type of conic. 

I. Rotation (Pages 690−693) 

The general equation of a conic whose axes are rotated so that 

they are not parallel to either the x-axis or the y-axis contains 

a(n) xy-term . 


How to rotate the 

coordinate axes to 

eliminate the xy-term in 

equations of conics 

To eliminate this term, you can use a procedure called rotation 

of axes , whose objective is to rotate the x- and y-axes 

until they are parallel to the axes of the conic. 

The general second-degree equation 

2 

2 

Ax + Bxy + Cy + Dx + Ey + F = 0 can be rewritten as 

2 

2 

A ′( 

x′ 

) + C′ 

( y′ 

) + D′ 

x′ 

+ E′ 

y′ 

+ F ′ = 0 by rotating the 

coordinate axes through an angle θ, where 

cot 2θ = (A − C)/B . 

The coefficients of the new equation are obtained by making the 

substitutions x = x′ cos θ − y′ sin θ and 

y = x′ sin θ + y′ cos θ . 

II. Invariants Under Rotation (Pages 694−695) 

Invariant under rotation means . . . that a term or 

quantity in the equation of a conic remains the same during a 

rotation of the coordinate axes through an angle θ. 


How to use the 

discriminant to classify 

conics 




The rotation of the coordinate axes through an angle θ that 

2 

2 

transforms the equation Ax + Bxy + Cy + Dx + Ey + F = 0 

2 

2 

into the form A ′( 

x′ 

) + C′ 

( y′ 

) + D′ 

x′ 

+ E′ 

y′ 

+ F ′ = 0 has the 

following rotation invariants: 

1) F = F ′ 

2) A + C = A′ + C ′ 

3) B 2 − 4AC = (B′) 2 − 4A′C ′ 

2 

The graph of the equation Ax + Bxy + Cy + Dx + Ey + F = 0 

is, except in degenerate cases, determined by its discriminant as 

follows: 

1) Ellipse or circle if: B 2 − 4AC < 0 

2) Parabola if : B 2 − 4AC = 0 

3) Hyperbola if: B 2 − 4AC > 0 

2 

Example 1: Classify the graph of the following conic: 

2x 

2 

Parabola 

2 

+ 12xy 

+ 18y 

− 3y 

− 5 = 0 

III. Systems of Quadratic Equations (Page 696) 

To find the points of intersection of two conics, . . . use 

either the elimination method or the substitution method. 


How to solve systems of 

quadratic equations 

Example 2: Solve the following system of quadratic equations: 

⎪⎧ 

2 2 

4x 

+ 4y 

− 36 = 0 

⎨ 

2 

⎪⎩ x − 3y 

− 6x 

+ 9 = 0 

(0, 3) and (3, 0) 


Page(s) 

Exercises 



Section 9.5 Parametric Equations 159 

Section 9.5 

Parametric Equations 

Objective: In this lesson you learned how to evaluate sets of parametric 

equations for given values of the parameter and graph curves 

that are represented by sets of parametric equations and how 

to rewrite sets of parametric equations as single rectangular 

equations and find sets of parametric equations for graphs. 

Course Number 

Instructor 

Date 



Parameter A third variable t introduced into equations involving x and y that relates 

the position of a point to time. 

I. Plane Curves (Page 699) 

If f and g are continuous functions of t on an interval I, the set of 

ordered pairs (f(t), g(t)) is a(n) plane curve C. The 

equations given by x = f (t) 

and y = g(t) are parametric 


How to evaluate sets of 

parametric equations for 

given values of the 

parameter 

equations for C, and t is the parameter . 

II. Sketching a Plane Curve (Pages 700−701) 

One way to sketch a curve represented by a pair of parametric 

equations is to plot points in the xy-plane . Each set 

of coordinates (x, y) is determined from a value chosen for the 

parameter t . By plotting the resulting points in the 

order of increasing values of t, you trace the curve in a specific 

direction, called the orientation of the curve. 


How to graph curves that 

are represented by sets of 

parametric equations 

Example 1: Sketch the curve described by the parametric 

2 + 

equations x = t − 3 and y = t 1, −1≤ 

t ≤ 3 . 

10 

y 

6 

2 

x 

-5 -3 -1 1 3 5 

-2 

-6 

-10 




Another way to display a curve represented by a pair of 

parametric equations is to use a graphing utility. To do so, . . . 

begin by setting the graphing utility to parametric mode, and 

enter the set of parametric equations as X 1T and Y 1T . When 

choosing a viewing window, you must set not only minimum 

and maximum values of x and y but also minimum and 

maximum values of t, the parameter. 

III. Eliminating the Parameter (Pages 702−703) 

Eliminating the parameter is the process of . . . finding a 

rectangular equation (in x and y) that has the same graph as a set 

of parametric equations. 


How to rewrite sets of 

parametric equations as 

single rectangular 

equations by eliminating 

the parameter 

Describe the process used to eliminate the parameter from a set 

of parametric equations. 

Start with the set of parametric equations. Solve for t in one 

equation. Then substitute for t in the second parametric equation. 

Simplify. The resulting equation is a rectangular equation. 

When converting equations from parametric to rectangular form, 

it may be necessary to alter . . . the domain of the 

rectangular equation so that its graph matches the graph of the 

parametric equations. 

To eliminate the parameter in equations involving trigonometric 

functions, try using the identities . . . 

sin 2 θ + cos 2 θ = 1 or sec 2 θ − tan 2 θ = 1 

IV. Finding Parametric Equations for a Graph 

(Page 703) 

Describe how to find a set of parametric equations for a given 

graph. 


How to find sets of 

parametric equations for 

graphs 



Page(s) 

Exercises 



Section 9.6 Polar Coordinates 161 

Section 9.6 

Polar Coordinates 

Objective: In this lesson you learned how to plot points in the polar 

coordinate system and convert equations from rectangular to 

polar form and vice versa. 

Course Number 

Instructor 

Date 


To form the polar coordinate system in the plane, fix a point O, 

called the pole or origin , and construct 

from O an initial ray called the polar axis . Then each 

point P in the plane can be assigned polar coordinates (r, θ) 

as follows: 


How to plot points and 

find multiple 

representations of points 

in the polar coordinate 

system 

1) r = directed distance from O to P 

2) θ = directed angle, counterclockwise from polar axis to 

the segment from O to P 

In the polar coordinate system, points do not have a unique 

representation. For instance, the point (r, θ) can be represented 

as (r, θ ± 2nπ) or (− r, θ ± (2n + 1)π) , 

where n is any integer. Moreover, the pole is represented by 

(0, θ), where θ is any angle . 

Example 1: Plot the point (r, θ) = (− 2, 11π/4) on the polar 

coordinate system. 

π/2 y 

π 

x0 

3π/2 

Example 2: Find another polar representation of the point 

(4, π/6). 

Answers will vary. One such point is (− 4, 7π/6). 




II. Coordinate Conversion (Pages 708−709) 

The polar coordinates (r, θ) are related to the rectangular 

coordinates (x, y) as follows . . . 


How to convert points 

from rectangular to polar 

form and vice versa 

x = r cos θ 

y = r sin θ 

tan θ = y/x r 2 = x 2 + y 2 

Example 3: Convert the polar coordinates (3, 3π/2) to 

rectangular coordinates. 

(0, − 3) 

III. Equation Conversion (Page 710) 

To convert a rectangular equation to polar form, . . . 

simply replace x by r cos θ and y by r sin θ, and simplify. 


How to convert equations 

from rectangular to polar 

form and vice versa 

Example 4: Find the rectangular equation corresponding to the 

− 5 

polar equation r = . 

sinθ 

y = − 5 

π/2 y 

π/2 y 

π/2 y 

π 

x0 

π 

x0 

π 

x0 

3π/2 


Page(s) 

Exercises 

3π/2 

3π/2 



Section 9.7 Graphs of Polar Equations 163 

Section 9.7 Graphs of Polar Equations 

Objective: In this lesson you learned how to graph polar equations. 


Example 1: Use point plotting to sketch the graph of the polar 

equation r = 3cosθ 

. 

π/2 y 

Course Number 

Instructor 

Date 


How to graph polar 

equations by point 

plotting 

π 

x0 

3π/2 

The graph of the polar equation r = f (θ ) can be rewritten in 

parametric form, using t as a parameter, as follows: 

x = f(t) cos t and y = f(t) sin t 

II. Symmetry (Pages 714−715) 

The graph of a polar equation is symmetric with respect to the 

following if the given substitution yields an equivalent equation. 

Substitution 

1) The line θ = π/2: Replace (r, θ) by (r, π − θ) or (− r, − θ) 


How to use symmetry as 

a sketching aid 

2) The polar axis: Replace (r, θ) by (r, − θ) or (− r, π − θ) 

3) The pole: Replace (r, θ) by (r, π + θ) or (− r, θ) 

Example 2: Describe the symmetry of the polar equation 

r = 2(1 

− sinθ 

) . 

Symmetric with respect to the line θ = π/2 

III. Zeros and Maximum r-Values (Pages 716−717) 

Two additional aids to sketching graphs of polar equations are . . . 

knowing the θ-values for which | r | is maximum and knowing 

the θ-values for which r = 0. 


How to use zeros and 

maximum r-values as 

sketching aids 




Example 3: Describe the zeros and maximum r-values of the 

polar equation r = 5cos 2θ 

The maximum value of | r | is | r | = 5 when θ = 0, 

π/2, π, 3π/2; the zeros of r occur at θ = π/4, 3π/4, 

5π/4, 7π/4 

IV. Special Polar Graphs (Pages 718−719) 

List the general equations that yield each of the following types 

of special polar graphs: 


How to recognize special 

polar graphs 

Limaçons: r = a ± b cos θ, r = a ± b sin θ, a > 0, b > 0 

Rose curves: 

r = a cos nθ, r = a sin nθ, n ≥ 2; n petals if n is odd; 2n petals if n is even 

Circles: 

r = a cos θ, r = a sin θ 

Lemniscates: 

r 2 = a 2 sin 2θ; r 2 = a 2 cos 2θ 

π/2 y 

π/2 y 

π/2 y 

π 

x0 

π 

x0 

π 

x0 

3π/2 

π/2 y 

3π/2 

π/2 y 

3π/2 

π/2 y 

π 

x0 

π 

x0 

π 

x0 

3π/2 


Page(s) 

Exercises 

3π/2 

3π/2 



Section 9.8 Polar Equations of Conics 165 

Section 9.8 Polar Equations of Conics 

Objective: In this lesson you learned how to write conics in terms of 

eccentricity and to write equations of conics in polar form. 

Course Number 

Instructor 

Date 

I. Alternative Definition of Conics (Page 722) 

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 conic . The constant 

ratio is the eccentricity of the conic and is 

denoted by e. Moreover, the conic is an ellipse if e < 1 , 

a parabola if e = 1 , and a hyperbola if e > 1 . 


How to define conics in 

terms of eccentricities 

For each type of conic, the focus is at the pole . 

II. Polar Equations of Conics (Pages 722−724) 

The graph of the polar equation r = ep/(1 + e cos θ) 

is a conic with a vertical directrix to the right of the pole, where 

e > 0 is the eccentricity and | p | is the distance between the focus 

(pole) and the directrix. 


How to write and graph 

equations of conics in 

polar form 

The graph of the polar equation r = ep/(1 − e cos θ) 

is a conic with a vertical directrix to the left of the pole, where 

e > 0 is the eccentricity and | p | is the distance between the focus 


The graph of the polar equation r = ep/(1 + e sin θ) 

is a conic with a horizontal directrix above the pole, where e > 0 

is the eccentricity and | p | is the distance between the focus 


The graph of the polar equation r = ep/(1 − e sin θ) 

is a conic with a horizontal directrix below the pole, where e > 0 




is the eccentricity and | p | is the distance between the focus 


Example 1: Identify the type of conic from the polar equation 

36 

r = 

, and describe its orientation. 

10 + 12sinθ 

Hyperbola with a horizontal directrix above the 

pole 

III. Applications (Page 725) 

Describe a real-life application of polar equations of conics. 



How to use equations of 

conics in polar form to 

model real-life problems 

π/2 y 

π/2 y 

π/2 y 

π 

x0 

π 

x0 

π 

x0 

3π/2 

π/2 y 

3π/2 

π/2 y 

3π/2 

π/2 y 

π 

x0 

π 

x0 

π 

x0 

3π/2 


Page(s) 

Exercises 

3π/2 

3π/2

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