How do you find the center and radius of the circle #X^2-8x+16+y^2+10y+25=81#?

1 Answer
Douglas K.
Jan 22, 2017

The given equation can be converted to the form #(x - h)^2 + (y - k)^2 = r^2# by observing the perfect squares within the equation. #(h,k)# is the center and r is the radius.

Explanation:

Given: #x^2 - 8x + 16 + y^2 + 10y + 25 = 81" [1]"#

Please observe that: #x^2 - 8x + 16 = (x - 4)^2# this makes equation [1] become:

#(x - 4)^2 + y^2 + 10y + 25 = 81" [2]"#

The y terms are, also, a perfect square: #y^2 + 10y + 25 = (y - -5)^2#

#(x - 4)^2 + (y - -5)^2 = 81" [3]"#

Finally, the constant term, #81 = 9^2#:

#(x - 4)^2 + (y - -5)^2 = 9^2#

The center is the point #(4,-5)# and the radius is 9.

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