What is the general form of the equation of the given circle with center a ?

the equation of circle is x²  + y² + 6x - 24y + 128 = 0

Step-by-step explanation:

To find the general form of the equation of the circle with center A (-3, 12) and radius r=5,  we simply use this formula;

General Equation of a circle is   (x - a)²  +  (y-b)² = r²

(a,b) are the two point at the center of the circle which are (-3, 12)

which implies a = -3 and b = 12

r is the radius of the circle which is given as 5 from the above diagram

to get r², we simply square 5 and so r²  = 25

We can now plug in the values of our variables into the equation

(x - a)²  +  (y-b)² = r²

(x - [-3])²   +   ( y  -  12)²  =   5²

(x +3)²   +   ( y  -  12)²  =   5²

we will expand all the brackets

x² +  6x +  9  + y² -24y + 144 = 25

x² +6x + y² -24y + 153 = 25

Take 25 to the left hand side of the equation

x² +6x + y² -24y + 153  -  25 = 0

x² +6x + y² -24y + 128 = 0

Rearranging the equation to give us a standard form of the equation of the circle, we have;

x² + y² + 6x - 24y  + 128 =0

{Note:  (x + 3)²  =  (x+3)(x+3) = x² + 6x + 9 and  

(y - 12)²  = (y -12)(y-12) = y² - 24y + 144}

Therefore the general form of the equation  of a circle with center(-3, 12) and radius 5 is x² + y² + 6x - 24y  + 128 =0

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