Question

complete the square to rewrite the equation of the circle in standard form. what is the equation of the circle in standard form? ( x^2 + y^2 + 4x - 10y + 20 = 0 ) ( (x - 2)^2 + (y + 5)^2 = 9 ) ( (x + 2)^2 + (y - 5)^2 = 9 ) ( (x - 4)^2 + (y + 10)^2 = 20 ) ( (x + 4)^2 + (y - 10)^2 = 20 )

Explanation:

Step1: Group x and y terms

Group the \(x\)-terms and \(y\)-terms together: \(x^{2}+4x + y^{2}-10y=-20\)

Step2: Complete the square for x

For \(x^{2}+4x\), take half of 4 (which is 2), square it (\(2^{2} = 4\)), and add to both sides: \(x^{2}+4x + 4+y^{2}-10y=-20 + 4\)
This becomes \((x + 2)^{2}+y^{2}-10y=-16\)

Step3: Complete the square for y

For \(y^{2}-10y\), take half of - 10 (which is - 5), square it (\((-5)^{2}=25\)), and add to both sides: \((x + 2)^{2}+y^{2}-10y + 25=-16+25\)
This simplifies to \((x + 2)^{2}+(y - 5)^{2}=9\)

Answer:

\((x + 2)^{2}+(y - 5)^{2}=9\)