Algebra II

Chapter 8: Conic Sections

Cheat Sheet• In chapter 8 you are allowed a “cheat sheet”

• You are to bring in a tissue box, that has not been opened, and cover it with paper.• You are allowed to write anything on this box that you so

choose.• You may use it on your quiz and chapter test.

• When we are done with Chapter 8, you must give the tissue box to me.

• It is your decision to do this, you may not have any other form a “cheat sheet”

8.1: Midpoint and Distance Formulas

• Find the midpoint of a segment on the coordinate plane

• Find the distance between two points on the coordinate plane

The Midpoint Formula

• The midpoint is the point in the middle of a segment• Definition: M is the midpoint of PQ if M is

between P and Q and PM = MQ.

• Formula:•

Example 1:

• Find the midpoint of each line segment with endpoints at the given coordinates:• (12, 7) and (-2, 11)

• (-8, -3) and (10, 9)

• (4, 15) and (10, 1)

• (-3, -3) and (3, 3)

“Curveball Problem” Example 2:

• Segment MN has a midpoint P. If M has coordinates (14, -3) and P has coordinates (-8, 6), what are the coordinates of N?

• Circle R has a diameter ST. If R has coordinates (-4, -8) and S has coordinates (1, 4), what are the coordinates of T?

“Curveball Problem” Example 2:

• Circle Q has a diameter AB. If A is at (-3,-5) and the center is at (2, 3), find the coordinates of the B.

The Distance Formula

• Distance is always a positive number

• You can find distance using the Pythagorean Theorem or using a formula derived from it

• Formula:•

Example 3:

• Find the distance between each pair of points with the given coordinates• (3, 7) and (-1, 4)

• (-2, -10) and (10, -5)

• (6, -6) and (-2, 0)

Example 4:

• Rectangle ABCD has vertices A(1, 4), B(3, 1), C(-3, -2), and D(-5, 1). Find the perimeter and area of ABCD

• Circle R has diameter ST with endpoints S(4, 5) and T(-2, -3). What are the circumference and are of the circle? (Round to two decimal places)

Summary:

• Learn the midpoint and distance formulas

• Be able to answer any question that may involve them.

• Questions?

8.2: Parabolas

• Write equations of parabola in standard form and vertex form

• Graph parabolas

Equations of Parabolas

• Standard Form• y = ax2 + bx + c

• Vertex Form• y = a(x – h)2 + k

Example 1:

• Write y = 3x2 + 24x + 50 in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

Example 1:

• Write y = -x2 – 2x + 3 in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

Graph Parabola

• You must always graph:• Vertex• Axis of Symmetry• Five points on the graph (this is to get

the shape)• Focus: point in which all points in a

parabola are equidistant• Directrix: line that the parabola will

never cross

Concept Summary (pg 422)Form of Equation y = a(x – h)2 + k x = a(y – k)2 + h

Vertex (h, k) (h, k)

Axis of Symmetry x = h y = k

Focus

Directrix

Direction of Opening

upward if a > 0downward if a < 0

right if a > 0left if a < 0

Example

Example 2:

• Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola• y = x2 + 6x – 4

• x = y2 – 8y + 6

Example 2:

• Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola• y = 8x – 2x2 + 10

• x = -y2 – 4y – 1

Example 3:

• Graph:y = ½(x – 1)2 + 2

• Graph:x = -2(y + 1)2 - 3

Classwork/Homework

• Workbook– Section 8.1

• 1, 3, 5, 11, 17, 19, 21, 31, 32

– Section 8.2• 1 – 6 (all)

8.3: Circles

• Write equations of circles

• Graph circles

Circle

• A circle is the set of all point in a plane that are equidistant from a given point in the plane, called the center.

• Equation of a circle:• (x – h)2 + (y – k)2 = r2

r = radius

center

(h, k)

Example One:

• Write an equation for the circle that satisfies each set of conditions:• Center (8, -3), Radius 6

• Center (5, -6), Radius 4

Example One:

• Write an equation for the circle that satisfies each set of conditions:• Center (-5, 2) passes through (-9, 6)

• Center (7, 7) passes through (12, 9)

Example One:

• Write an equation for the circle that satisfies each set of conditions:• Endpoints of a diameter are (-4, -2) and (8,

4)

• Endpoints of a diameter are (-4, 3) and (6, -8)

Graph circles

• Make sure the equation is in standard form

• Graph the center

• Use the length of the radius to graph four points on the circle (up, down, left, right)

• Connect the dots to create the circle

Example Two:• Find the center and radius of the

circle given the equation. Then graph the circle• (x – 3)2 + y2 = 9

Example Two:• Find the center and radius of the

circle given the equation. Then graph the circle• (x – 1)2 + (y + 3)2 = 25

Example Two:• Find the center and radius of the circle

given the equation. Then graph the circle• x2 + y2 – 10x + 8y + 16 = 0

Example Two:• Find the center and radius of the circle

given the equation. Then graph the circle• x2 + y2 – 4x + 6y = 12

Classwork/Homework

• Workbook• Lesson 8.3

• 1 – 13 (all)

Homework Answers: Workbook 8.3

1. (x + 4)2 + (y – 2)2 = 642. x2 + y2 = 163. (x + ¼)2 + (y + )2 = 504. (x – 2.5)2 + (y – 4.2)2 = 0.815. (x + 1)2 + (y + 7)2 = 56. (x + 9)2 + (y + 12)2 = 747. (x + 6)2 + (y – 5)2 = 258. (-3, 0); r = 49. (0, 0); r = 210. (-1, -3); r = 611. (1, -2); r = 412. (3, 0); r = 313. (-1, -3); r = 3

3

Homework Review

8.4: Ellipses

• Write equations of ellipses

• Graph ellipses

Ellipse• An ellipse is like an oval.

• Every ellipse has two axes of symmetry• Called the major axis and the minor axis

• The axes intersect at the center of the ellipse• The major axis is bigger than the minor axis• We use c2 = a2 – b2 to find c

• a is always greater b

• The equation is always equal to 1

Ellipses Chart (pg 434)

Standard Form of Equation

Center (h,k) (h,k)

Direction of Major Axis

Horizontal Vertical

Foci (h + c, k), (h – c, k)

(h, k + c), (h, k – c)

Length of Major Axis

2a units 2a units

Length of Minor Axis

2b units 2b units

2 2

2 2

x - h y - k+ =1

a b

2 2

2 2

y - k x - h+ =1

a b

Example One:

Graph the ellipse 2 2x - 2 y + 5

+ =14 1

Your Turn:

Graph the ellipse 2 2x + 2 y - 5

+ =181 16

Example Two:

Graph the ellipse 2 2y - 4 x - 2

+ =164 4

Your Turn:

Graph the ellipse 2 2y - 2 x - 4

+ =136 9

Example Three:

Write the equation of the ellipse in the graph:

Your Turn:

Write the equation of the ellipse in the graph:

Example Four:

Write the equation of the ellipse in the graph:

Your Turn:

Write the equation of the ellipse in the graph:

Standard Form

Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation: x2 + 4y2 + 24y = -32

Standard Form

Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation: 9x2 + 6y2 – 36x + 12y = 12

Classwork

Hyperbolas Chart

Standard Form of Equation

Direction of Transverse Axis

Horizontal Vertical

Foci (h + c, 0), (h - c, 0)

(0, h + c), (0, h - c)

Vertices (h + a, 0), (h - a, 0)

(0, h + a), (0, h - a)

Length of Transverse Axis

2a units 2a units

Length of Conjugate Axis

2b units 2b units

Equations of Asymptotes

2 2

2 2

x - h y - k- =1

a b

2 2

2 2

y - k x - h- =1

a b

by - k = ± (x - h)

aa

y - k = ± (x - h)b

Example One:

Graph the hyperbola

22 y x - = 1

4 9

Your Turn:

Graph the hyperbola

22 y x - = 1

1 4

Example Two:

Graph the hyperbola

2 2x - 4 y + 2

- 9 16

Your Turn:

Graph the hyperbola

2 2x - 3 y + 5

- 9 25

Example Three:

Write the equation of the hyperbola in the graph:

Your Turn:

Write the equation of the hyperbola in the graph:

Standard Form:

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with equation

4x2 – 9y2 – 32x – 18y + 19 = 0

Standard Form:

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with equation

x2 – y2 + 6x + 10y – 17 = 0

8.6 Conic Sections

The equation of any conic section can be written in the general quadratic equation:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0where A, B, and C ≠ 0

If you are given an equation in this general form, you can complete the square to write the equation in one of the standard forms you have already learned.

Standard Forms (you already know )

Conic Section

Standard Form of Equation

Parabola y = a(x – h)2 + k x = a(y – k)2 + h

Circle (x – h)2 + (y – k)2 = r2

Ellipse

Hyperbola

2 2

2 2

x - h y - k+ =1

a b

2 2

2 2

y - k x - h+ =1

a b

2 2

2 2

x - h y - k- =1

a b 2 2

2 2

y - k x - h- =1

a b

Identifying Conic Sections

Relationship of A and C Type of Conic Section

Only x2 or y2 Parabola

Same number in front of x2 and y2

Circle

Different number in front of x2 and y2 with plus sign

Ellipse

Different number in front of x2 and y2 with plus sign or minus sign

Hyperbola

Example One:

Write each equation in standard form. Then state whether the graph of the equation is a parabola, circle, ellipse, and hyperbola.y = x2 + 4x + 1

x2 + y2 = 4x + 2

y2 – 2x2 – 16 = 0

x2 + 4y2 + 2x – 24y + 33 = 0

Example Two:

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, and hyperbola.x2 + 2y2 + 6x – 20y + 53 = 0

x2 + y2 – 4x – 14y + 29 = 0

3y2 + x – 24y + 46 = 0

6x2 – 5y2 + 24x + 20y – 56 = 0

Your Turn:

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, and hyperbola.x2 + y2 – 6x + 4y + 3 = 0

6x2 – 60x – y + 161 = 0

x2 – 4y2 – 16x + 24y – 36 = 0

x2 + 2y2 + 8x + 4y + 2 = 0

Classwork/Homework

WorkbookPage 56

1 – 3, 8 – 10

Page 571 – 12