Given the equation of a circle in the general form as [tex]\( A x^2 + B y^2 + C x + D y + E = 0 \)[/tex] where [tex]\( A = B \neq 0 \)[/tex], and knowing that the radius of the circle is 3 units and the center lies on the y-axis, let's deduce the correct set of values for [tex]\( A, B, C, D, \)[/tex] and [tex]\( E \)[/tex].
Step 1: Properties of the Circle's Equation
1. Since [tex]\( A = B \)[/tex] and they are not zero, this simplifies the form to [tex]\( x \)[/tex] and [tex]\( y \)[/tex] having the same coefficient.
2. The circle is centered on the [tex]\( y \)[/tex]-axis, which implies the [tex]\( x \)[/tex]-coordinate of the center, [tex]\( h \)[/tex], is 0.
3. The radius [tex]\( R \)[/tex] is given as 3 units, so [tex]\( R^2 = 9 \)[/tex].
Step 2: Center and Radius Conditions
Using the standard circle equation form [tex]\((x - h)^2 + (y - k)^2 = R^2\)[/tex] where center [tex]\((h, k)\)[/tex]:
1. Since the circle is centered on the y-axis, [tex]\( h = 0 \)[/tex], the center becomes [tex]\( (0, k) \)[/tex].
2. The equation transforms to [tex]\( x^2 + (y - k)^2 = 9 \)[/tex].
Step 3: Converting to General Form
Expanding [tex]\( x^2 + y^2 - 2ky + k^2 = 9 \)[/tex]:
1. Combine like terms: [tex]\( x^2 + y^2 - 2ky + (k^2 - 9) = 0 \)[/tex].
Here, matching with [tex]\( A x^2 + B y^2 + C x + D y + E = 0 \)[/tex]:
- [tex]\( C = 0 \)[/tex] (since there is no linear term in [tex]\( x \)[/tex]),
- [tex]\( D = -2k \)[/tex],
- [tex]\( E = k^2 - 9 \)[/tex].
Step 4: Checking the Given Options
Let's analyze each option:
A. [tex]\( A = 0, B = 0, C = 2, D = 2, E = 3 \)[/tex]
- This invalidates [tex]\( A = B \neq 0 \)[/tex].
B. [tex]\( A = 1, B = 1, C = 8, D = 0, E = 9 \)[/tex]
- [tex]\( C \neq 0 \)[/tex] contradicts the fact that there should be no [tex]\( x \)[/tex]-linear term.
C. [tex]\( A = 1, B = 1, C = 0, D = -8, E = 7 \)[/tex]
- Here, [tex]\( C = 0 \)[/tex] matches.
- [tex]\( D = -2k \Rightarrow k = 4 \)[/tex].
- Checking [tex]\( E: k^2 - 9 = 4^2 - 9 = 16 - 9 = 7 \)[/tex], which matches.
D. [tex]\( A = 1, B = 1, C = -8, D = 0, E = 0 \)[/tex]
- [tex]\( C \neq 0 \)[/tex] contradicts the fact that there should be no [tex]\( x \)[/tex]-linear term.
E. [tex]\( A = 1, B = 1, C = 8, D = 8, E = 3 \)[/tex]
- [tex]\( C \neq 0 \)[/tex] contradicts the fact that there should be no [tex]\( x \)[/tex]-linear term.
Hence, the set of values that correspond to the given circle is:
[tex]\[ \boxed{(C) \ A=1, \ B=1, \ C=0, \ D=-8, \ E=7.} \][/tex]
