Equations of Circles H2-12

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• Equations of Circles and tangents

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This photograph was taken 216 miles above Earth.
From this altitude, it is easy to see
the curvature of the horizon. Facts about circles
can help us understand details about Earth.
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Definitions

• circle all points that are the same distance from
  the center of the circle.
• A circle with center C is called circle C, or ?C.
• A radius is a line segment from the center to
  the circle.
• A diameter is a line segment or chord that passes
  through the center.

radius
center
diameter
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• The word radius (plural radii) is also used to
  denote the length of a radius
  
• The word diameter is also used to denote the
  length of a diameter
• Note that the diameter of a circle is twice its
  radius.

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Equation of a Circle
The center of a circle is given by (h, k) The
radius of a circle is given by r The equation of
a circle in standard form is (x h)2 (y k)2
r2
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The standard form of the equation of a circle
with its center at the origin is
r is the radius of the circle so if we take the
square root of the right hand side, we'll know
how big the radius is.
Notice that both the x and y terms are squared.
Linear equations dont have either the x or y
terms squared. Parabolas have only the x term
was squared (or only the y term, but NOT both).
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A1.
Center at (0, 0)
This is r2 so r 3
The center of the circle is at the origin and the
radius is 3. Let's graph this circle.
Count out 3 in all directions since that is the
radius
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If the center of the circle is NOT at the origin
then the equation for the standard form of a
circle looks like this
The center of the circle is at (h, k).
This is r2 so r 4
Find the center and radius and graph this circle.
The center of the circle is at (h, k) which is
(3,1).
The radius is 4
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If you take the equation of a circle in standard
form for example
This is r2 so r 2
(x - (-2))
Remember center is at (h, k) with (x - h) and
(y - k) since the x is plus something and not
minus, (x 2) can be written as (x - (-2))
You can find the center and radius easily. The
center is at (-2, 4) and the radius is 2.
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Identify the center and radius and sketch the
graph
Remember the center values end up being the
opposite sign of what is with the x and y and the
right hand side is the radius squared.
So the center is at (-4, 3) and the radius is 5.
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Now lets reverse it, find the equation when
given center and radius
B1. Find an equation of the circle with center at
(0, 0) and radius 7.
Let's sub in center and radius values in the
standard form
0
7
0
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B2. Find an equation of the circle with center at
(-2, 5) and radius 6
Subbing in the values in standard form we have
-2
5
6
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Practice

• Write the standard equation of the circle
• B3. Center (0, 4) Radius of 1

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Practice

• Write the standard equation of the circle
• B4. Center (-7, 0) Radius of 100

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Finding the Equation of a Circle
Circle A The center is (16, 10) The radius
is 10 The equation is (x 16)2 (y 10)2
100
Note the scale on this graph is by 2s
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Finding the Equation of a Circle

• Circle B
• The center is (4, 20)
• The radius is 10
• The equation is
• (x 4)2 (y 20)2 100

Note the scale on this graph is by 2s
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Finding the Equation of a Circle
Circle O The center is (0, 0) The radius is
12 The equation is x 2 y 2 144
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But what if it was not in standard form but
multiplied out
Moving everything to one side, putting the xs
and ys together and combining like terms we'd
have
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Now, subtract the 16 on both sides and complete
the square for both the x's and y's to get in
standard form.
Move constant to the other side
Group x terms and a place to complete the square
Group y terms and a place to complete the square
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4
16
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Complete the square
Write factored and wahlah! back in standard form.
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Put all the xs and ys together
constant on the end
We have to complete the square on both the x's
and y's to get in standard form.
Move constant to the other side
Group x terms and a place to complete the square
Group y terms and a place to complete the square
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9
4
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Write factored for standard form.
So the center is at (-3, 2) and the radius is 4.
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• 1. If DE is a tangent line,
• What is the angle x?
• Because DE is a tangent line
• It must be perpendicular to the radius/diameter
  at the point of tangency.
• Angle D must be a right angle
• x 180 38 90 52

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• 2. If AB is tangent to circle C,
• Find the radius.
• Tangent is perpendicular
• Use Pythagorean Theorem
• C2 B2 A2
• (x8)2 122 x2
• x2 16x 64 144 x2
• 16x 80
• x 5

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• Is ML tangent to Circle N
• at L? Explain.
• If ML is tangent, than it will
• create a right angle at the
• point of tangency, and right triangle NLM
• We can use the Pythagorean theorem to check if it
  is a right triangle
• 72 242 252
• 49 576 625
• 625 625 , ?NLM is a right triangle
• ?L is 90 degrees
• LM is perpendicular to the radius at L, so
• LM is tangent to Circle N