To solve the problem, we need to start by recalling the geometric properties and equations associated with a circle.
1. General Form of the Circle
The general form of a circle’s equation in Cartesian coordinates is:
[tex]\[ Ax^2 + By^2 + Cx + Dy + E = 0 \][/tex]
Given in the problem, [tex]\( A = B \neq 0 \)[/tex].
2. Standard Form of the Circle
The standard form of a circle with center at [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
3. Converting General Form to Standard Form
Since [tex]\(A = B = 1\)[/tex],
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0 \][/tex]
We need to complete the square to convert this into the standard form.
4. Determine Center and Radius Conditions
Given:
- The radius [tex]\( r = 3 \)[/tex].
- The center of the circle lies on the [tex]\( y \)[/tex]-axis, so [tex]\( h = 0 \)[/tex].
Hence, the center is at [tex]\( (0, k) \)[/tex].
5. Equation in Standard Form
For a circle centered at [tex]\( (0, k) \)[/tex], the equation in standard form becomes:
[tex]\[ (x - 0)^2 + (y - k)^2 = 3^2 \][/tex]
[tex]\[ x^2 + (y - k)^2 = 9 \][/tex]
Expanding this:
[tex]\[ x^2 + y^2 - 2yk + k^2 = 9 \][/tex]
6. Match Coefficients to General Form
Compare this with the general form:
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0 \][/tex]
Since there is no [tex]\( x \)[/tex]-term in the standard form, we have [tex]\( C = 0 \)[/tex].
For the [tex]\( y \)[/tex]-term, we have [tex]\( -2yk = Dy \)[/tex] implying [tex]\( D = -2k \)[/tex].
For the constants: [tex]\( k^2 - 9 = E \)[/tex].
7. Results Analysis
Look at various options and compare:
- Option A: [tex]\( A = 0, B = 0, C = 2, D = 2, E = 3 \)[/tex] – Incorrect, as [tex]\( A \)[/tex] and [tex]\( B \)[/tex] should both be 1.
- Option B: [tex]\( A = 1, B = 1, C = 8, D = 0, E = 9 \)[/tex] – The values [tex]\( C = 8\)[/tex] and [tex]\(D = 0 \)[/tex] do not match our derived results.
- Option C: [tex]\( A = 1, B = 1, C = 0, D = 8, E = ? \)[/tex] – Checking these values:
The coefficient [tex]\( C = 0 \)[/tex] is consistent. For [tex]\( D = 8\)[/tex], [tex]\( -2k = 8\)[/tex], giving [tex]\( k = -4 \)[/tex]. Thus, we calculate [tex]\( E = k^2 - 9 \rightarrow (-4)^2 - 9 = 16 - 9 = 7 \)[/tex]. Hence, [tex]\( E = -7\)[/tex].
- Option D: [tex]\( A = 1, B = 1, C = -8, D = 0, E = 0 \)[/tex] – Incorrect calculation from base requirement.
- Option E: [tex]\( A = 1, B = 1, C = 8, D = 8, E = 3 \)[/tex] – Incorrect base components.
8. Conclusion
The correct values, tested for option C, ensure that:
[tex]\[ A = 1, B = 1, C = 0, D = 8 \][/tex] and the derived value [tex]\( E = 0\)[/tex].
Thus, the correct set is:
[tex]\[ \boxed{C: A = 1, B=1, C=0, D=8, E=0} \][/tex]
