Unit 13.1

UNIT 13.1 CONIC SECTIONUNIT 13.1 CONIC SECTIONBASICSBASICS

Warm UpSolve for y.

1. x2 + y2 = 1

2. 4x2 – 9y2 = 1

Recognize conic sections as intersections of planes and cones.

Use the distance and midpoint formulas to solve problems.

Objectives

conic section

Vocabulary

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.

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.

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.

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.

4x2 + 25y2 = 100


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

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).


Example 1B Continued

Graph each equation on a graphing calculator. Identify each conic section. Then describe the center and intercepts.

x2 + y2 = 49


Check It Out! Example 1a



Check It Out! Example 1a Continued

9x2 + 25y2 = 225


Check It Out! Example 1b



Check It Out! Example 1b Continued

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

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


y = – x2 12

y = – x2 12

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

Example 2B


y2 – x2 = 9

y2 = 9 + x2 Add x2 to both sides.

Take the square root of both sides.

Example 2B Continued


The graph is a hyperbola that opens vertically with vertices at (0, ±3).

and

Graph each equation on a graphing calculator. Identify each conic section. Then describe the vertices and the direction that the graph opens.


2y2 = x

Check It Out! Example 2a


Check It Out! Example 2a Continued


x2 – y2 = 16

Check It Out! Example 2b


Check It Out! Example 2b Continued

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.

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

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

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

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

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

Lesson Quiz: Part IGraph 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

Lesson Quiz: Part II

5. Find the center and radius of a circle that has a diameter with endpoints (3, 7) and (–2, –5).

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Unit 13.1