The equation of a circle is given by:
[tex]\[
x^2 + y^2 + Cx + Dy + E = 0
\][/tex]
First, convert this to the standard form of a circle [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
1. Determine the center (h, k):
- By completing the square, we express the equation in this form:
- [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex] where:
- [tex]\(h = -\frac{C}{2}\)[/tex]
- [tex]\(k = -\frac{D}{2}\)[/tex]
2. Determine the radius [tex]\(r\)[/tex]:
- The radius [tex]\(r\)[/tex] is given by:
[tex]\[
r = \sqrt{h^2 + k^2 - E}
\][/tex]
From this, we can rearrange to solve for [tex]\(E\)[/tex]:
[tex]\[
E = h^2 + k^2 - r^2
\][/tex]
3. Decrease the radius:
- When the radius is decreased, say the new radius is [tex]\(r_{\text{new}}\)[/tex], the equation becomes:
[tex]\[
E_{\text{new}} = h^2 + k^2 - r_{\text{new}}^2
\][/tex]
4. Analyze the change in [tex]\(E\)[/tex]:
- Since [tex]\(h\)[/tex] and [tex]\(k\)[/tex] remain unchanged (center does not change), the values [tex]\(h^2 + k^2\)[/tex] are also constants.
- If [tex]\(r\)[/tex] is decreased, [tex]\(r_{\text{new}}\)[/tex] is smaller than [tex]\(r\)[/tex], making [tex]\(r_{\text{new}}^2\)[/tex] smaller than [tex]\(r^2\)[/tex].
- Therefore, [tex]\(E_{\text{new}} = h^2 + k^2 - r_{\text{new}}^2\)[/tex] is greater than [tex]\(h^2 + k^2 -r^2\)[/tex].
In conclusion, the coefficients [tex]\(C\)[/tex] and [tex]\(D\)[/tex] which are associated with the center coordinates [tex]\((h, k)\)[/tex] do not change. However, the coefficient [tex]\(E\)[/tex] increases as the radius decreases.
The correct answer is:
E. [tex]\(C\)[/tex] and [tex]\(D\)[/tex] are unchanged, but [tex]\(E\)[/tex] increases.
