Introduction to Conic Sections 10-1 Introduction to Conic Sections Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Solve for y. 1. x2 + y2 = 1 2. 4x2 – 9y2 = 1

Objectives Recognize conic sections as intersections of planes and cones. Use the distance and midpoint formulas to solve problems.

Vocabulary conic section

IF we slice the cone, parallel to the base, what do we get? A Circle!

IF we slice the cone at an angle, what do we get now? An Ellipse!

IF we just take a slice from the lateral face of the cone, what do we get? A Parabola!

Finally, let’s take a slice from the lateral face, perpendicular to the base A Hyperbola!

These shapes are called Conic Sections These shapes are called Conic Sections. We can use Algebra to describe the equations and graphs of these shapes.

In Chapter 5, you studied the parabola 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.

A circle is defined by its center and its radius 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. When you take the square root of both sides of an equation, remember that you must include the positive and negative roots. Remember!

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.

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.

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 Subtract 4x2 from both sides. y2 = 100 – 4x2 25 Divide both sides by 25. Take the square root of both sides.

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.

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.

Check It Out! Example 1a Continued Step 2 Use two equations to see the complete graph. Check Use a table to confirm the intercepts.

Check It Out! Example 1b 9x2 + 25y2 = 225 Step 1 Solve for y so that the expression can be used in a graphing calculator.

Check It Out! Example 1b Continued Step 2 Use two equations to see the complete graph. Check Use a table to confirm the intercepts.

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

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 = – x2 1 2 Step 1 Solve for y so that the expression can be used in a graphing calculator. y = – x2 1 2

Step 2 Use the equation to see the complete graph. Example 2A Continued Step 2 Use the equation to see the complete graph. y = – x2 1 2 The graph is a parabola with vertex (0, 0) that opens downward.

Example 2B y2 – x2 = 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.

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).

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.

Check It Out! Example 2a Continued Step 2 Use two equations to see the complete graph.

Check It Out! Example 2b x2 – y2 = 16 Step 1 Solve for y so that the expression can be used in a graphing calculator.

Check It Out! Example 2b Continued Step 2 Use two equations to see the complete graph.

Every conic section can be defined in terms of distances 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

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). ( , ) = (2.5, –2) 5 + 0 2 4 – 8

 r = ( 5 – 2.5)2 + (4 – (–2))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. r = ( 5 – 2.5)2 + (4 – (–2))2  The radius is the same using (5, 4).

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.

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. 

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

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