Converting Circle Equation to Center-Radius Form
The equation written in the image appears to be:
\[ x^2 + y^2 + 2x - 4y - 11 = 0 \]
This looks to be the equation of a circle in the general form, which can be converted into the center-radius form by completing the square for both x and y.
The general equation for a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where (h,k) is the center of the circle and r is its radius.
Let's complete the square for the given equation:
1. Group x and y terms together and add and subtract the necessary constants inside the parenthesis to form a perfect square trinomial:
\[ (x^2 + 2x + 1) + (y^2 - 4y + 4) - 1 - 4 - 11 = 0 \]
2. Complete the square by adding and subtracting the right constants inside each parenthesis:
\[ (x^2 + 2x + 1) + (y^2 - 4y + 4) = 1 + 4 + 11 \]
3. Rewrite as squares and simplify the right side of the equation:
\[ (x + 1)^2 + (y - 2)^2 = 16 \]
Thus, the center-radius form of the given circle's equation is:
\[ (x + 1)^2 + (y - 2)^2 = 4^2 \]
So the center of the circle is (-1, 2) and the radius is 4.
