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Algebra (all content)

Course: Algebra (all content) > Unit 17

Lesson 4: Expanded equation of a circle
Math
Algebra (all content)
Conic sections
Expanded equation of a circle
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Circle equation review

Google Classroom
Review the standard and expanded forms of circle equations, and solve problems concerning them.

What is the standard equation of a circle?

left parenthesis, x, minus, start color #11accd, h, end color #11accd, right parenthesis, squared, plus, left parenthesis, y, minus, start color #ca337c, k, end color #ca337c, right parenthesis, squared, equals, start color #e07d10, r, end color #e07d10, squared
This is the general standard equation for the circle centered at left parenthesis, start color #11accd, h, end color #11accd, comma, start color #ca337c, k, end color #ca337c, right parenthesis with radius start color #e07d10, r, end color #e07d10.
Circles can also be given in expanded form, which is simply the result of expanding the binomial squares in the standard form and combining like terms.
For example, the equation of the circle centered at left parenthesis, start color #11accd, 1, end color #11accd, comma, start color #ca337c, 2, end color #ca337c, right parenthesis with radius start color #e07d10, 3, end color #e07d10 is left parenthesis, x, minus, start color #11accd, 1, end color #11accd, right parenthesis, squared, plus, left parenthesis, y, minus, start color #ca337c, 2, end color #ca337c, right parenthesis, squared, equals, start color #e07d10, 3, end color #e07d10, squared. This is its expanded equation:
(x−1)2+(y−2)2=32(x2−2x+1)+(y2−4y+4)=9x2+y2−2x−4y−4=0\begin{aligned} (x-\blueD 1)^2+(y-\maroonD 2)^2&=\goldD 3^2 \\\\ (x^2-2x+1)+(y^2-4y+4)&=9 \\\\ x^2+y^2-2x-4y-4&=0 \end{aligned}
Want to learn more about circle equations? Check out this video.

Practice set 1: Using the standard equation of circles

Problem 1.1
  • Current
left parenthesis, x, plus, 4, right parenthesis, squared, plus, left parenthesis, y, minus, 6, right parenthesis, squared, equals, 48
What is the center of the circle?
left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis
What is its radius?
If necessary, round your answer to two decimal places.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units

Want to try more problems like this? Check out this exercise and this exercise.

Practice set 2: Writing circle equations

Problem 2.1
  • Current
A circle has a radius of square root of, 13, end square root units and is centered at left parenthesis, minus, 9, point, 3, comma, 4, point, 1, right parenthesis.
Write the equation of this circle.

Want to try more problems like this? Check out this exercise.

Practice set 3: Using the expanded equation of circles

To interpret the expanded equation of a circle, we should rewrite it in standard form using the method of "completing the square."
Consider, for example, the process of rewriting the expanded equation x, squared, plus, y, squared, plus, 18, x, plus, 14, y, plus, 105, equals, 0 in standard form:
x2+y2+18x+14y+105=0x2+y2+18x+14y=−105(x2+18x)+(y2+14y)=−105(x2+18x+81)+(y2+14y+49)=−105+81+49(x+9)2+(y+7)2=25(x−(−9))2+(y−(−7))2=52\begin{aligned} x^2+y^2+18x+14y+105&=0 \\\\ x^2+y^2+18x+14y&=-105 \\\\ (x^2+18x)+(y^2+14y)&=-105 \\\\ (x^2+18x\redD{+81})+(y^2+14y\blueD{+49})&=-105\redD{+81}\blueD{+49} \\\\ (x+\redD9)^2+(y+\blueD7)^2&=25 \\\\ (x-(-9))^2+(y-(-7))^2&=5^2 \end{aligned}
Now we can tell that the center of the circle is left parenthesis, minus, 9, comma, minus, 7, right parenthesis and the radius is 5.
Problem 3.1
  • Current
x, squared, plus, y, squared, minus, 10, x, minus, 16, y, plus, 53, equals, 0
What is the center of this circle?
left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis
What is the radius of this circle?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units

Want to try more problems like this? Check out this exercise and this exercise.

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