Question

select the correct answer.
a circle is described by the equation ( x^2 + y^2 - 6x + 8y = 0 ) what are the coordinates of the center of the circle and the length of its radius?
a. ( (3, -4) ), 5 units
b. ( (-3, 4) ), 25 units
c. ( (-3, 4) ), 5 units
d. ( (3, -4) ), 25 units

Explanation:

Step1: Rewrite equation, group variables

$x^2 -6x + y^2 +8y = 0$

Step2: Complete the square for x

$x^2-6x+9 = (x-3)^2$, add 9 to both sides:
$(x-3)^2 + y^2+8y = 9$

Step3: Complete the square for y

$y^2+8y+16 = (y+4)^2$, add 16 to both sides:
$(x-3)^2 + (y+4)^2 = 9+16$

Step4: Simplify to circle form

$(x-3)^2 + (y+4)^2 = 25 = 5^2$

Step5: Identify center and radius

Center $(h,k)=(3,-4)$, radius $r=5$ units

Answer:

A. (3,-4), 5 units