Conic Sections

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Conic Sections

• Circles

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Objective

• To learn the relationship between the center and
  the radius of a circle and the equation of the
  circle.

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Conics
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Conic Sections

• Circle
• Parabola
• Ellipse
• Hyperbola

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For every conic

• We will look at
• the equation
• the graph
• putting an equation in standard conic form

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Equation of a Circle

• A circle is the set of all points in a plane that
  are a fixed distance from a fixed point.
• Equation comes from the Pythagorean Theorem

(x, y)
(h,k)
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Example

• Find an equation of the circle with center (4,
  3) and radius 2.
• (x 4)2 (y 3)2 22

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• Graph (x 2)2 (y 6)2 4.

• To graph you need a location and a size.
• In this case, find the center and the radius.
• C (2, -6) and r 2

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• Graph (x 2)2 (y 6)2 4.
• Now we will graph this on the Ti-Nspire
• But we must solve for y and graph both the
• top and bottom halves.

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Putting it in standard form

• x2 y2 10x 4y 21 0
• x2 10x y2 4y -21
• x2 10x 25 y2 4y 4 -21 25 4
• (x 5)2 (y 2)2 8
• C (-5, 2), r

gather your variables and isolate the constant
complete the square for the xs and the ys
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How about another one?

• x2 y2 4y 5 0
• x2 y2 4y 5
• x2 y2 4y 4 5 4
• x2 (y 2)2 9
• C (0, 2), r 3

gather your variables and isolate the constant
complete the square for the ys
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Last one..

• x2 y2 2x 2y 8 0
• x2 2x y2 2y -8
• x2 2x 1 y2 2y 1 -8 1 1
• (x 1)2 (y 1)2 -6
• Wait! Cant be!
• (x 1)2 will be positive and (y 1)2 will be
  positive and that can not add to be a negative!
• This is not a circle!

gather your variables and isolate the constant
complete the square for the xs and ys
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Homework

• Page 410 1 24 (all),
• 31 38 (all)