Find the Center and Radius x^2+y^2-6x+16y+57=0

Problem

x^2+y^2−6*x+16*y+57=0

Solution

1. Group the terms by variable and move the constant term to the right side of the equation.

(x^2−6*x)+(y^2+16*y)=−57

2. Complete the square for the x terms by adding ((−6)/2)^2=9 to both sides.

(x^2−6*x+9)+(y^2+16*y)=−57+9

3. Complete the square for the y terms by adding (16/2)^2=64 to both sides.

(x^2−6*x+9)+(y^2+16*y+64)=−57+9+64

4. Simplify the right side and write the left side as squared binomials.

(x−3)^2+(y+8)^2=16

5. Identify the center (h,k) and the radius r by comparing the equation to the standard form (x−h)^2+(y−k)^2=r^2

h=3

k=−8

r^2=16⇒r=4

Final Answer

Center: *(3,−8), Radius: *4

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