10-1
10-1 Introduction
Introduction to
to Conic
Conic Sections
Sections
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
10-1 Introduction to Conic Sections
Warm Up
Solve for y.
1. x2 + y2 = 1
2. 4x2 – 9y2 = 1
Holt Algebra 2
10-1 Introduction to Conic Sections
Objectives
Recognize conic sections as
intersections of planes and cones.
Use the distance and midpoint formulas
to solve problems.
Holt Algebra 2
10-1 Introduction to Conic Sections
Vocabulary
conic section
Holt Algebra 2
10-1 Introduction to Conic Sections
In Chapter 5, you studied the
parabola. The parabola is one
of a family of curves called
conic sections. Conic
sections are formed by the
intersection of a double right
cone and a plane. There are
four types of conic sections:
circles, ellipses, hyperbolas,
and parabolas.
Although the parabolas you studied in Chapter 5 are
functions, most conic sections are not. This means
that you often must use two functions to graph a
conic section on a calculator.
Holt Algebra 2
10-1 Introduction to Conic Sections
A circle is defined by its center and its radius. An
ellipse, an elongated shape similar to a circle,
has two perpendicular axes of different lengths.
Remember!
When you take the square root of both sides of
an equation, remember that you must include
the positive and negative roots.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 1A: Graphing Circles and Ellipses on a
Calculator
Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
center and intercepts.
(x – 1)2 + (y – 1)2 = 1
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
(y – 1)2 = 1 – (x – 1)2
Subtract (x – 1)2 from both sides.
Take square root of both sides.
Then add 1 to both sides.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 1A Continued
Step 2 Use two equations to see the complete graph.
Use a square window on your
graphing calculator for an accurate
graph. The graphs meet and form a
complete circle, even though it might
not appear that way on the calculator.
The graph is a circle with center (1, 1)
and intercepts (1,0) and (0, 1).
Check Use a table to
confirm the intercepts.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 1B: Graphing Circles and Ellipses on a
Calculator
4x2 + 25y2 = 100
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
25y2 = 100 – 4x2
2
y =
100 – 4x2
25
Subtract 4x2 from both sides.
Divide both sides by 25.
Take the square root of both
sides.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 1B Continued
Step 2 Use two equations to see the complete graph.
Use a square window on your graphing
calculator for an accurate graph. The
graphs meet and form a complete ellipse,
even though it might not appear that way
on the calculator.
The graph is an ellipse with center (0, 0)
and intercepts (±5, 0) and (0, ±2).
Check Use a table to
confirm the intercepts.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 1a
Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
center and intercepts.
x2 + y2 = 49
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 1a Continued
Step 2 Use two equations to see the complete graph.
Check Use a table to
confirm the intercepts.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 1b
9x2 + 25y2 = 225
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 1b Continued
Step 2 Use two equations to see the complete graph.
Check Use a table to
confirm the intercepts.
Holt Algebra 2
10-1 Introduction to Conic Sections
A parabola is a single curve, whereas a hyperbola
has two congruent branches. The equation of a
2
2
parabola usually contains either an x term or a y
term, but not both. The equations of the other
conics will usually contain both x2 and y2 terms.
Helpful Hint
Because hyperbolas contain two curves that open
in opposite directions, classify them as opening
horizontally, vertically, or neither.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 2A: Graphing Parabolas and Hyperbolas on
a Calculator
Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
vertices and the direction that the graph opens.
y=–
1
2
x2
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
y=–
Holt Algebra 2
1
2
x2
10-1 Introduction to Conic Sections
Example 2A Continued
Step 2 Use the equation to see the complete graph.
y=–
1
2
x2
The graph is a parabola
with vertex (0, 0) that
opens downward.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 2B
2
2
y –x = 9
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
y2 = 9 + x2
Add x2 to both sides.
Take the square root of both
sides.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 2B Continued
Step 2 Use two equations to see the complete graph.
and
The graph is a hyperbola
that opens vertically
with vertices at (0, ±3).
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 2a
Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
vertices and the direction that the graph opens.
2y2 = x
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 2a Continued
Step 2 Use two equations to see the complete graph.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 2b
x2 – y2 = 16
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 2b Continued
Step 2 Use two equations to see the complete graph.
Holt Algebra 2
10-1 Introduction to Conic Sections
Every conic section can be defined in terms of
distances. You can use the Midpoint and Distance
Formulas to find the center and radius of a circle.
Holt Algebra 2
10-1 Introduction to Conic Sections
Because a diameter must
pass through the center of
a circle, the midpoint of a
diameter is the center of
the circle. The radius of a
circle is the distance from
the center to any point on
the circle and equal to half
the diameter.
Helpful Hint
The midpoint formula uses averages. You can
think of xM as the average of the x-values and yM
as the average of the y-values.
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 3: Finding the Center and Radius of a Circle
Find the center and radius of a circle that has
a diameter with endpoints (5, 4) and (0, –8).
Step 1 Find the center of the circle.
Use the Midpoint Formula with the endpoints
(5, 4) and (0, –8).
(
5+0 4–8
2 ,
2 )
Holt Algebra 2
= (2.5, –2)
10-1 Introduction to Conic Sections
Example 3 Continued
Step 2 Find the radius.
Use the Distance Formula with (2.5, –2) and
(0, –8)
The radius of the circle is 6.5
Check Use the other endpoint (5, 4) and the
center (2.5, –2). The radius should equal 6.5 for
any point on the circle.
r = ( 5 – 2.5)2 + (4 – (–2))2 
The radius is the same using (5, 4).
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 3
Find the center and radius of a circle that has
a diameter with endpoints (2, 6) and (14, 22).
Step 1 Find the center of the circle.
Holt Algebra 2
10-1 Introduction to Conic Sections
Check It Out! Example 3 Continued
Step 2 Find the radius.
Check Use the other endpoint (2, 6) and the
center (8, 14). The radius should equal 10 for any
point on the circle.

Holt Algebra 2
10-1 Introduction to Conic Sections
Lesson Quiz: Part I
Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
center and intercepts for circles and ellipses, or
the vertices and direction that the graph opens for
parabolas and hyperbolas.
1. x2 – 16y2 = 16
2. 4x2 + 49y2 = 196
3. x = 6y2
4. x2 + y2 = 0.25
Holt Algebra 2
10-1 Introduction to Conic Sections
Lesson Quiz: Part II
5. Find the center and radius of a circle that
has a diameter with endpoints (3, 7) and
(–2, –5).
Holt Algebra 2