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![Page 1: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/1.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
![Page 2: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/2.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Warm UpSolve for y.
1. x2 + y2 = 1
2. 4x2 – 9y2 = 1
![Page 3: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/3.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Recognize conic sections as intersections of planes and cones.
Use the distance and midpoint formulas to solve problems.
Objectives
![Page 4: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/4.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
conic section
Vocabulary
![Page 5: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/5.jpg)
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.
![Page 6: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/6.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
When you take the square root of both sides of an equation, remember that you must include the positive and negative roots.
Remember!
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.
![Page 7: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/7.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Graph each equation on a graphing calculator. Identify each conic section. Then describe the center and intercepts.
Example 1A: Graphing Circles and Ellipses on a Calculator
(x – 1)2 + (y – 1)2 = 1
Step 1 Solve for y so that the expression can be used in a graphing calculator.
Subtract (x – 1)2 from both sides.(y – 1)2 = 1 – (x – 1)2
Take square root of both sides.
Then add 1 to both sides.
![Page 8: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/8.jpg)
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.
![Page 9: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/9.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
4x2 + 25y2 = 100
Step 1 Solve for y so that the expression can be used in a graphing calculator.
Subtract 4x2 from both sides.25y2 = 100 – 4x2
Take the square root of both sides.
y2 = 100 – 4x2
25Divide both sides by 25.
Example 1B: Graphing Circles and Ellipses on a Calculator
![Page 10: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/10.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
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.
Example 1B Continued
![Page 11: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/11.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
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.
Check It Out! Example 1a
![Page 12: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/12.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Step 2 Use two equations to see the complete graph.
Check Use a table to confirm the intercepts.
Check It Out! Example 1a Continued
![Page 13: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/13.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
9x2 + 25y2 = 225
Step 1 Solve for y so that the expression can be used in a graphing calculator.
Check It Out! Example 1b
![Page 14: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/14.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Step 2 Use two equations to see the complete graph.
Check Use a table to confirm the intercepts.
Check It Out! Example 1b Continued
![Page 15: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/15.jpg)
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 parabola usually contains either an x2 term or a y2 term, but not both. The equations of the other conics will usually contain both x2 and y2 terms.
Because hyperbolas contain two curves that open in opposite directions, classify them as opening horizontally, vertically, or neither.
Helpful Hint
![Page 16: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/16.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Graph each equation on a graphing calculator. Identify each conic section. Then describe the vertices and the direction that the graph opens.
Example 2A: Graphing Parabolas and Hyperbolas on a Calculator
Step 1 Solve for y so that the expression can be used in a graphing calculator.
y = – x2 12
y = – x2 12
![Page 17: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/17.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 2A Continued
Step 2 Use the equation to see the complete graph.
The graph is a parabola with vertex (0, 0) that opens downward.
y = – x2 12
![Page 18: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/18.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 2B
Step 1 Solve for y so that the expression can be used in a graphing calculator.
y2 – x2 = 9
y2 = 9 + x2 Add x2 to both sides.
Take the square root of both sides.
![Page 19: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/19.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Example 2B Continued
Step 2 Use two equations to see the complete graph.
The graph is a hyperbola that opens vertically with vertices at (0, ±3).
and
![Page 20: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/20.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Graph each equation on a graphing calculator. Identify each conic section. Then describe the vertices and the direction that the graph opens.
Step 1 Solve for y so that the expression can be used in a graphing calculator.
2y2 = x
Check It Out! Example 2a
![Page 21: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/21.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Step 2 Use two equations to see the complete graph.
Check It Out! Example 2a Continued
![Page 22: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/22.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Step 1 Solve for y so that the expression can be used in a graphing calculator.
x2 – y2 = 16
Check It Out! Example 2b
![Page 23: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/23.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Step 2 Use two equations to see the complete graph.
Check It Out! Example 2b Continued
![Page 24: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/24.jpg)
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.
![Page 25: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/25.jpg)
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.
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.
Helpful Hint
![Page 26: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/26.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
Find the center and radius of a circle that has a diameter with endpoints (5, 4) and (0, –8).
Example 3: Finding the Center and Radius of a Circle
Step 1 Find the center of the circle.
Use the Midpoint Formula with the endpoints (5, 4) and (0, –8).
( , ) = (2.5, –2)5 + 0 2
4 – 8 2
![Page 27: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/27.jpg)
Holt Algebra 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.
The radius is the same using (5, 4).
r = ( 5 – 2.5)2 + (4 – (–2))2
![Page 28: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/28.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
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.
Check It Out! Example 3
![Page 29: Holt Algebra 2 10-1 Introduction to Conic Sections 10-1 Introduction to Conic Sections Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dda5503460f94ad0945/html5/thumbnails/29.jpg)
Holt Algebra 2
10-1 Introduction to Conic Sections
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.
Check It Out! Example 3 Continued