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College Algebra – MT 150
Day #4: Section 3.6 – Introduction to Circles
Today, we will change our focus towards non-linear equations, coming in the form
of circles. In the process, we want to accomplish four main goals.
1. We want to know and understand the standard form for a circle.
Standard form for a circle: _______________________________________
Where ‘r’ is the _____________
and (h,k) are the coordinates of the _____________.
2. We want to be able to write the standard form of circles, using information
that we are given.
a. If you are told that the circle is ‘TANGENT TO THE X-AXIS’, then use
the |𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒| as the radius.
b. If you are told that the circle is ‘TANGENT TO THE Y-AXIS’, then use
the |𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒| as the radius.
c. If you are told the ‘ENDPOINTS OF THE DIAMETER,’ then use the
DISTANCE FORMULA to find the length of the diameter and take ½ of
that to plug in for the radius.
DISTANCE FORMULA:
Relationship between diameter and radius:
3. We want to be able to write the formula for a circle by looking at a graph of
a circle.
4. We need to be able to ‘COMPLETE THE SQUARE’ when we sometimes want
to find the center and radius of a circle, based on a given equation. The
process of ‘COMPLETING THE SQUARE” involves taking ½ of the xcoordinate and then squaring the number. Add that number to both sides
of the equation and then write a binomial.
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Examples:
Instructions: Find the standard form of the equation for the circle:
1. Center (-4, 3) and radius 5
2. Center (7, -9) and radius 3
3. Center at ORIGIN; radius 3
4. Center (√5, √3) and radius 4
5. Center at (7, 2) and tangent to x-axis
6. Center at (-3, 8) and passes
through (-4, 9)
7. Center at (4, 8) and passes through (1, 9)
8. Center at the origin and
passes through (6, -7)
9. Endpoints of a diameter are (-8, 6) and (1, 11)
10. Endpoints of a diameter are (-7, -4) and (-5, 7)
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11. Based on the graph of the circle below, find the standard form of the
equation for the circle.
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12. Based on the graph of the circle below, find the standard form of the
equation for the circle
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Instructions: Based on each given equation, find the CENTER and RADIUS of
each circle.
1. 𝑥 2 + 𝑦 2 = 36
2. 𝑥 2 + (𝑦 − 8)2 = 9
3. (𝑥 + 5)2 + (𝑦 + 4)2 = 4
4. (𝑥 − 7)2 + (𝑦 − 2)2 = 16
5. 𝑥 2 + 𝑦 2 − 4𝑥 + 4𝑦 − 8 = 0
6. 𝑥 2 + 𝑦 2 + 10𝑦 + 9 = 0
6. (𝑥 − 1)2 + 𝑦 2 = 9
HOMEWORK ASSIGNMENT:
I.
P. 239-240 (2-12 even, 18, 25, 26, 28)
II.
On page 240, do (30, 32, 34, 36, 40, 42, 46), but IGNORE the instructions
in the book and only do what it says below:
Based on each given equation, find the CENTER and RADIUS of each circle.