Circles -- Section 2.4 --

 The Standard Equation of a Circle*: (x - h)2 + (y - k)2 = r2 General Form of the Equation of a Circle*: x2 + y2 + cx + dy + e = 0 *These names may vary from book to book.

 Find the equation in general form of a circle with the center at (0,0) and radius of  6.
 Solve Step r = 6 and (h, k) = (0, 0) (0,0) is the origin (x - h)2 + (y - k)2 = r2 Use Standard Equation form (x - 0)2 + (y - 0)2 = 62 Simplify x2 + y2  = 36 Set equal to zero x2 + y2  - 36 = 0 General Form
 Find the equation in general form of a circle with the center at (0,0) and radius of  4.
 [Solution]
 Find the equation in general form of a circle with the center at (-3, 2) and radius of  6.
 Solve Step r = 6 and (h, k) = (-3, 2) Plug in the values (x - h)2 + (y - k)2 = r2 Use Standard Equation form (x - (-3))2 + (y - 2)2 = 62 Simplify (x + 3)2 + (y - 2)2 = 36 Put in general form x2 + 6x + 9 + y2  - 4y + 4 = 36 Multiply out x2 + y2 + 6x - 4y + 13 = 36 Combine like terms and reorder terms x2 + y2 + 6x - 4y  - 23 = 0 Set equal to zero for general form General Form: x2 + y2 + cx + dy + e = 0
 Find the equation in general form of a circle with the center at (4, -1) and radius of  5.
 [Solution]
 Given:  x2 +  y2 + 10x - 3y  + 3 = 0 Put in the Standard Equation form.
 Solve Step x2 +  y2 + 10x - 3y  + 3 = 0 Group like terms x2 + 10x +  y2 - 3y  = -3 Subtract 3 from both sides x2 + 10x +  52 + y2 - 3y  + (3/2)2 = -3 + 52 + (3/2)2 * Take half the middle term and square it. Note half of 3 is 3/2 (x + 5)2 + (x - 3/2)2  =  -3 + 25 + 9/4 Complete the squares!! Add those fractions... (x + 5)2 + (x - 3/2)2  = 97/4 Standard Equation form: (x - h)2 + (y - k)2 = r2 Multiply (x + 5)2 + (x - 3/2)2  = 97/4  to verify The center is (-5, 3/2)  Radius is *Click here for completing the square help with this step.
 Given:  x2 +  y2 - 5x + 4y  - 2 = 0 Put in the Standard Equation form.
 [Solution]
 Graph the equation:  x2  + y2  - 4y  - 21 = 0
 Solve Step x2  + y2  - 4y  - 21 = 0 Put in the Standard Equation form x2  +  y2 - 4y  = 21 Add 21 to both sides x2  +  y2  - 4y + 4 = 21 + 4 Take half the middle term and square it. x2  +  (y  - 2)2  = 25 Standard Equation form (x - h)2 + (y - k)2 = r2 The center is (0, 2) and r = 5 Locate the center (0,2) Then count 5 units along the x and y direction. Draw a beautiful circle. Graph the equation:  x2 + 6x + y2  - 7  = 0 [Solution]

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