Questions: Write into graphing form of a circle to determine the center x radius x^2+y^2+8x-6y+16=0

Transcript text: Write into graphing form of a circle to determine the center $x$ radius \[ x^{2}+y^{2}+8 x-6 y+16=\varnothing \]

Solution

Solution Steps

Step 1: Rearrange the equation

Start by grouping the $ x $-terms and $ y $-terms together: \[ x^{2} + 8x + y^{2} - 6y + 16 = 0 \]

Step 2: Move the constant term to the other side

Subtract 16 from both sides to isolate the $ x $- and $ y $-terms: \[ x^{2} + 8x + y^{2} - 6y = -16 \]

Step 3: Complete the square for the $ x $-terms

Take half of the coefficient of $ x $ (which is 8), square it, and add it to both sides: \[ x^{2} + 8x + 16 + y^{2} - 6y = -16 + 16 \] This simplifies to: \[ (x + 4)^{2} + y^{2} - 6y = 0 \]

Step 4: Complete the square for the $ y $-terms

Take half of the coefficient of $ y $ (which is -6), square it, and add it to both sides: \[ (x + 4)^{2} + y^{2} - 6y + 9 = 0 + 9 \] This simplifies to: \[ (x + 4)^{2} + (y - 3)^{2} = 9 \]

Step 5: Identify the center and radius

The equation is now in the standard form of a circle: \[ (x - h)^{2} + (y - k)^{2} = r^{2} \] Comparing, we find: