To understand how the coefficients [tex]\( C \)[/tex], [tex]\( D \)[/tex], and [tex]\( E \)[/tex] are affected when the radius of a circle is decreased without changing the coordinates of the center point, let's start by analyzing the general equation of a circle in standard form and compare it with the given equation.
1. Standard Form of Circle Equation:
The equation of a circle with center [tex]\((a, b)\)[/tex] and radius [tex]\(r\)[/tex] is given by:
[tex]\[
(x - a)^2 + (y - b)^2 = r^2
\][/tex]
2. Expanding the Standard Form:
Let's expand [tex]\((x - a)^2 + (y - b)^2 = r^2\)[/tex]:
[tex]\[
x^2 - 2ax + a^2 + y^2 - 2by + b^2 = r^2
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - r^2) = 0
\][/tex]
3. Comparing with Given Equation:
The expanded form [tex]\(x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - r^2) = 0\)[/tex] looks similar to the given equation [tex]\(x^2 + y^2 + Cx + Dy + E = 0\)[/tex] where:
[tex]\[
C = -2a,\quad D = -2b,\quad \text{and} \quad E = a^2 + b^2 - r^2
\][/tex]
4. Effect of Changing the Radius:
- Coefficient [tex]\(C\)[/tex]: This coefficient depends on the [tex]\(x\)[/tex]-coordinate of the center [tex]\(a\)[/tex]. Since the center does not change, [tex]\(C\)[/tex] remains unchanged.
- Coefficient [tex]\(D\)[/tex]: Similarly to [tex]\(C\)[/tex], this coefficient depends on the [tex]\(y\)[/tex]-coordinate of the center [tex]\(b\)[/tex]. Since the center does not change, [tex]\(D\)[/tex] remains unchanged.
- Coefficient [tex]\(E\)[/tex]: This coefficient involves the radius [tex]\(r\)[/tex]. Specifically, [tex]\(E = a^2 + b^2 - r^2\)[/tex]. If the radius [tex]\(r\)[/tex] decreases, [tex]\(r^2\)[/tex] decreases, which in turn increases [tex]\(E\)[/tex].
Hence, the correct option that describes the changes to the coefficients is:
E. [tex]\(C\)[/tex] and [tex]\(D\)[/tex] are unchanged, but [tex]\(E\)[/tex] increases.
