Distance and Midpoint Formula - Welcome To Mr. Whitlow's ...



Distance and Midpoint Formula

§7.1

Distance Formula

[pic]

Example 1

Find the distance between the points (4, 4) and (-6, -2).

Example 2

Find the value of a to make the distance = 10 units given the points

(-7, 3) and (a, 11).

Example 3

Show that M(2, 4) is the midpoint of the segment joining

A(7, 2) and B(-3, 6).

Midpoint (Line Segment)

[pic]

Example 4

Find the center of a segment whose coordinates are

A(-2, 3) and B(8, -5).

Example 5

Circle x has a diameter MN. If M is at (-4, 2) and the center is (-6, 3), find the coordinates of N.

Pg 411, 2-34 even

Parabolas

§7.2

Conic Sections – any figure that can be formed by slicing a double cone.

Parabola – the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix.

Information about Parabolas

|Form of Equation |[pic] |[pic] |

|Axis of Symmetry |x = h |y = k |

|Vertex |(h, k) |(h, k) |

|Focus |[pic] |[pic] |

|Directrix |[pic] |[pic] |

|Direction of Opening |a(+): Up, a(-): Down |a(+): Right, a(-): Left |

|Length of Latus |[pic] |[pic] |

Example 1

Graph [pic]

Example 2

Graph [pic]

Pg 419, 6-12, 16-21,25,28

Parabolas

§7.2 (Day 2)

Writing Equations

Need a and vertex to write equation

Remember:

1. Focus, Directrix, and Latus all have a as part of their formula.

2. a and Latus are reciprocals of one another.

Example 3

Write the equation of the cross-section of a satellite dish with focus 2 units from the vertex and a latus 8 units long. Assume that the focus is at the origin and the parabola opens to the right.

Example 4

Write and equation for the parabola shown below.

Example 5

Focus = (-4, -2) and Directrix: x = -8

Pg 419, 30-43

Circles

§7.3

Circle – the set of all points in a plane that are equidistant from a given point in the plane, called the center.

Equation of a Circle

[pic]

Center: (h, k)

Radius: r

Example 1

Write an equation of a circle that has a radius 15 and a center

(-9, -6) and then graph.

Example 2

Find the center and radius of circle with equation [pic] and then graph.

Tangent – a line that intersects a circle at exactly one point.

Example 3

Write an equation of the circle that has its center at (2, -4) and is tangent to the x-axis.

Example 4

Write an equation of a circle if the endpoints of a diameter are at (1, 8) and (1, -6).

Pg 426, 1-49 odds

Ellipses

§7.4

Ellipse – the set of all points in a plane such that the sum off the distances from the foci is constant.

**An ellipse has 2 axis of symmetry**

[pic]

Standard Equations of Ellipses with center at origin

[pic]

***the largest value is always a***

Example 1

Find the coordinates of the foci and the lengths of the major and minor axes of an ellipse whose equation is [pic] and then graph.

Example 2

Find the coordinates of the foci and the lengths of the major and minor axes of an ellipse whose equation is [pic] and then graph.

Writing Equations

Need a and b

Example 3

Write the equation of the ellipse shown below.

Pg 436, 6, 8, 10, 14-16, 19, 20, 23-26, 35

Ellipses

§7.4 (Day 2)

Standard Equations of Ellipses with center (h, k)

[pic]

Example 4

Graph [pic]

Example 2

An equation of an ellipse is [pic]. Find the coordinates of the center, foci, the lengths of the major and minor axes, and then graph.

Pg 436, 7, 9, 11-13, 17, 18, 21, 22, 27-32, 34, 36

Hyperbolas

§7.5

Hyperbola – the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points, called the foci, is constant.

[pic]

Standard Equations of Hyperbolas with Center at Origin

[pic]

**when x is positive, the transverse axis is horizontal; when y is positive, the transverse axis is vertical.**

|Equation of |[pic] |[pic] |

|Hyperbola | | |

|Equation of |[pic] |[pic] |

|Asymptote | | |

|Transverse |Horizontal |Vertical |

|Axis | | |

*Positive value is where the transverse axis lies*

Example 1

A comet travels along a path that is one branch of a hyperbola whose equation is [pic]. Find the coordinates of the vertices, foci, equations of the asymptotes, and then draw the figure.

Example 2

Write an equation for the hyperbola [pic] in standard form and graph.

Example 3

Write an equation of a hyperbola with a foci at (0, 7) and (0, -7) if the length of the transverse axis is 6 units.

Remember, when writing equations we need a and b.

Pg 445, 5-8, 10, 11, 14-16, 19-22, 37

Hyperbolas

§7.5 (Day 2)

Standard Equations of Hyperbolas with Center at (h, k)

[pic]

**when x is positive, the transverse axis is horizontal; when y is positive, the transverse axis is vertical.**

Example 1

Draw the graph of [pic]

Example 2

Write the equation of the Hyperbola shown below.

Example 3

The graph of [pic] is a hyperbola.

Find the standard form of the equation, coordinates of the vertices and foci, the equations of the asymptotes and draw.

Pg 445, 9, 12, 13, 18, 23-25, 35, 36

Conic Sections

§7.6

Equation of a Conic Section: [pic]

|Conic Section |Standard Form of Equation |Relationship of |

| | |A and C |

|Parabola |[pic] |A = 0 or C = 0, not both |

|Circle |[pic] |A = C |

|Ellipse |[pic] |A and C have the same sign and [pic] |

|Hyperbola |[pic] |A and C have |

| | |opposite signs |

Example 1

Identify [pic] and graph.

Example 2

Identify [pic] and graph.

Example 3

Identify [pic] and graph.

Pg 453, 5-31, 41, 43 odd

Solving Quadratic Systems

§7.7

Example 1

Solve.

[pic]

[pic]

Algebra Graphing

2 Conic Sections

[pic]

Example 2

Solve

[pic]

[pic]

Algebra Graphing

Example 3

Solve algebraically.

[pic]

[pic]

Example 4

Solve by Graphing.

[pic]

[pic]

Pg 464, 7-37 odd

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