Question

The equation of a circle is x2 + y2 + Cx + Dy + E = 0. If the radius of the circle is decreased without changing the coordinates of the center point, how are the coefficients C, D, and E affected?

Answer 1

\[\begin{align*}
x^2+y^2+Cx+Dy+E&=0\\\\
x^2+Cx+\frac{C^2}{4}+y^2+Dy+\frac{D^2}{4}&=\frac{C^2}{4}+\frac{D^2}{4}-E\\\\
\left(x+\frac{C}{2}\right)^2+\left(y+\frac{D}{2}\right)^2&=\frac{C^2}{4}+\frac{D^2}{4}-E
\end{align*}\]
This is the equation of a circle with center \(\left(-\dfrac{C}{2},\,-\dfrac{D}{2}\right)\) and radius \(\dfrac{C^2}{4}+\dfrac{D^2}{4}-E\).

If the radius were to change without altering the center of the circle, you would have no change in C or D because the center directly depends on these coefficients.