Question

1. $x^{2}=-y^{2}-10x$
2. $x^{2}+y^{2}-9 = 8y$

write the equation of the circle in general form.
how do we do this? multiply the center - radius form out: $x^{2}+y^{2}+cx + dy+e = 0$

1. $(x + 3)^{2}+(y - 1)^{2}=4$
2. $(y + 8)^{2}+(x + 2)^{2}=72$

Explanation:

Step1: Expand \((x + 3)^2+(y - 1)^2=4\)

Use the formula \((a + b)^2=a^{2}+2ab + b^{2}\) and \((a - b)^2=a^{2}-2ab + b^{2}\).
\((x + 3)^2=x^{2}+6x + 9\), \((y - 1)^2=y^{2}-2y + 1\).
So \(x^{2}+6x + 9+y^{2}-2y + 1=4\).

Step2: Simplify the equation

Combine like - terms: \(x^{2}+y^{2}+6x-2y+9 + 1-4=0\).
\(x^{2}+y^{2}+6x-2y + 6=0\).

Step3: Expand \((y + 8)^2+(x + 2)^2=72\)

\((y + 8)^2=y^{2}+16y+64\), \((x + 2)^2=x^{2}+4x + 4\).
So \(y^{2}+16y+64+x^{2}+4x + 4=72\).

Step4: Simplify the equation

Combine like - terms: \(x^{2}+y^{2}+4x+16y+64 + 4-72=0\).
\(x^{2}+y^{2}+4x+16y-4=0\).

Answer:

1. \(x^{2}+y^{2}+6x-2y + 6=0\)
2. \(x^{2}+y^{2}+4x+16y-4=0\)