2.2 Circles, 2.3 Circles, Analyze

Find the center and the radius from the equation

A circle is the collection of points that are equidistant to a center point.  The distance is the radius denoted r.  The center is denoted (h, k).

The standard form of an equation of a circle is (x-h)^2+(y-k)^2=r^2 where r is the radius and   (h, k) is the center.

 

Example:  Find the center and the radius from the given equation.

(x-2)^2+(y-3)^2=16

Solution:  If you line up the standard form of a circle with the equation given you can determine the center and the radius.

In the binomial with x, the number after the minus sign is h.  Thus, h=2.  In the binomial with y, the number after the minus sign is k.  Thus, k=3.  The r^2 lines up with 16.  So r^2=16 which means r=4.

This is the equation of a circle with a center of (2, 3) and a radius of r=4 .

 

Example:  Find the center and the radius from the given equation.

(x+1)^2+(y-4)^2=25

Solution:  Notice that one of the binomials has a + instead of the – that is in the standard form of the circle.  Rewrite the addition as subtraction.

x+1 is the same as x-(-1).

The rewritten equation is as follows.

(x-(-1))^2+(y-4)^2=25

Then, if you line up the standard form of a circle with the equation given you can determine the center and the radius.

In the binomial with x, the number after the minus sign is h.  Thus, h=-1.  In the binomial with y, the number after the minus sign is k.  Thus, k=4.  The r^2 lines up with 25.  So r^2=25 which means r=5.

This is the equation of a circle with a center of (-1, 4) and a radius of r=5 .

 

Example:  Find the center and the radius from the given equation.

x^2+(y+7)^2=12

Solution:  Notice that one of the binomials has a + instead of the – that is in the standard form of the circle.  Rewrite the addition as subtraction.

y+7 is the same as y-(-7).

The other binomial doesn’t have any number added or subtracted.  We can rewrite this by subtracting zero.

  x is the same as x-0.

The rewritten equation is as follows.

(x-0)^2+(y-(-7))^2=12

Then, if you line up the standard form of a circle with the equation given you can determine the center and the radius.

In the binomial with x, the number after the minus sign is h.  Thus, h=0.  In the binomial with y, the number after the minus sign is k.  Thus, k=-7.  The r^2 lines up with 25.  So r^2=12 which means r=sqrt{12}=2sqrt{3}.

This is the equation of a circle with a center of (0, -7) and a radius of r=2sqrt{3} .