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ch 10.1 and 10.2.notebook 1 April 24, 2014 Chapter 10: Conic Sections
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Chapter 10: Conic Sections
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Warm Up
10.1
Solve for y
1. x2+y2=9 2. 9x225y=16
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Ch 10.1: Intro to Conic Sections
Content Objective: I will be able to recognize conic sections as intersections of planes and cones and use the distance and midpoint formulas to solve problems.
Vocabulary
Conic Sections: a plane figure formed by the intersection of a double right cone and a plane. The four types of conic sections are circles, ellipses, hyperbolas, and parabolas
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Studied parabolas as function, most conic sections are not functions.
i.e. you have to use more than one function to graph a conic section
Note:
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https://itunes.apple.com/us/podcast/a10.2aconicsections/id478153085?i=104294129&mt=2
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Graphing Circles and Ellipses on a Calculator
Ex 1:
A.
Graph each equation by solving for y. Identify each conic section. Then describe the center and intercepts.
Conic Section:
Center: ( , )
Xint(s) @: ( , ), ( , )
Yint(s) @: ( , ), ( , )
solve for y:
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Graph
Can check table for intercepts
notice the relation b/w intercepts and equation
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B.
Conic Section:
Center: ( , )
Xint(s) @: ( , ), ( , )
Yint(s) @: ( , ), ( , )
solve for y:
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notice the relation b/w intercepts and equation
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Ex 2:
Graphing Parabolas and Hyperbolas on a Calculator
Conic Section:
Center: ( , )
Xint(s) @: ( , ), ( , )
Yint(s) @: ( , ), ( , )
A. 3y2 = x
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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 y2 term, but not both. The equations of the other conics will usually contain both x2 and y2 terms
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Conic Section:
Center: ( , )
Xint(s) @: ( , ), ( , )
Yint(s) @: ( , ), ( , )
B) x2 – y2 = 4
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Because the diameter of a circle must pass through its center, the midpoint of that line is the center.
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Ex. 3 Find the center and the radius of the circle below.
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Find the slope of the line that connects each pair of points. 1. (5, 7) and (–1, 6) 2. (3, –4) and (–4, 3)
Find the distance between each pair of points. 3. (–2, 12) and (6, –3) 4. (1, 5) and (4, 1)
Warm Up
10.2
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10.2: Circles
Content Objective: I will be able to write an equation for a circle, graph a circle, and identify the circle’s radius and its center.
Vocabulary
Circle: A set of points equidistance (radius) from a point (center).
Tangent: a line in the same plane as the circle that intersects the circle at exactly one point.
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Ex 1: Write the equation of a circle with center (2,1) and radius r=5.
Notice that r2 and the center are visible in the equation of the circle
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Ex 2: Write the equation of the following circles:
Hint: use distance formula to find radius
Writing the Equation of a Circle
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Writing the Equation of a Tangent LineEx 3:
Write the equation of the line that is tangent to the circle 25 = x2 + y2 at the point (3, 4).Step 1: Identify the center and radius of the circle.
Step 2: Find the slope of the radius at the point of tangency and the slope of the tangent.
Step 3: Find the slopeintercept equation of the tangent
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Ex 4:
Try itWrite the equation of the line that is tangent to the circle 25 = (x – 1)2 + (y + 2)2 at the point (5, ‐5) .
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Completing the Square (Review)
Expand the binomial.A. (x + 3)2 =
Factor the trinomial.
B. x2 – 12x + 36
What term is needed to complete the square? C) a2 + 8a + ______
Ex 1:
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Complete the Square
A) x2 + 6x + 8 = 0
HINTS:1. Move the constant to the other side.2. Add the new number to BOTH sides.3. Factor.
____ + ____ = ____
____ + ____ + ____ = ____ + ____
( ____ + ____ )2 = ____
Ex 2:
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4x2 + 8x 16 = 0
HINTS:1. Move the constant to the other side.2. Divide everything by the coefficient of x2.3. Add the new number to BOTH sides.4. Factor.
____ + ____ = ____
____ + ____ = ____
____ + ____ + ____ = ____ + ____
( ____ + ____ )2 = ____
Ex 2:
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Every conic section may be written in the form
Ax2 + By2 + Cx + Dy + E = 0
Recall that standard form for a circle: (x h)2 + (y k)2 = r2
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Ex 3:
3x2 + 3y2 6x + 24y + 24 = 0(Hint: Divide everything by 3 first!)
Complete the Square
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Note: If your 'r' is negative you cannot take the square root and the eq has no sol.
(x+4)2 + (y3)2 = 36
This does not represent an equation of a circle
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Ex 4:
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10.3: Ellipses
Content Objective: I will be able to write the standard equation for an ellipse, graph an ellipse, and identify its center, vertices, covertices, and foci.
![[LEC]1.2 Conic Sections](https://static.documents.pub/doc/80x56/577d2eb01a28ab4e1eafbd09/lec12-conic-sections.jpg)
