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

What is the standard equation of a circle?

(xh)2+(yk)2=r2(x-\blueD h)^2+(y-\maroonD k)^2=\goldD r^2
This is the general standard equation for the circle centered at (h,k)(\blueD h, \maroonD k) with radius r\goldD r.
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 (1,2)(\blueD 1,\maroonD 2) with radius 3\goldD 3 is (x1)2+(y2)2=32(x-\blueD 1)^2+(y-\maroonD 2)^2=\goldD 3^2. This is its expanded equation:
(x1)2+(y2)2=32(x22x+1)+(y24y+4)=9x2+y22x4y4=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

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

Practice set 2: Writing circle equations

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 x2+y2+18x+14y+105=0x^2+y^2+18x+14y+105=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 (9,7)(-9,-7) and the radius is 55.
Want to try more problems like this? Check out this exercise and this exercise.
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