Answer :
To solve the given problem, we need to determine the coefficients [tex]\(A, B, C, D, E\)[/tex] in the equation of the circle [tex]\(A x^2 + B y^2 + C x + D y + E = 0\)[/tex] such that the circle has certain properties; namely, a radius of 3 units and its center lies on the [tex]\(y\)[/tex]-axis.
The center of the circle [tex]\((h, k)\)[/tex] can be found by recognizing that it can be derived from the general form. The given condition states that the center lies on the [tex]\(y\)[/tex]-axis, meaning [tex]\(h = 0\)[/tex], and the radius [tex]\(R = 3\)[/tex].
The standard form of the circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = R^2 \][/tex]
Since [tex]\(h = 0\)[/tex]:
[tex]\[ x^2 + (y - k)^2 = 9 \][/tex]
Expanding and rearranging this into the general form:
[tex]\[ x^2 + y^2 - 2ky + k^2 = 9 \][/tex]
We need to identify [tex]\(A, B, C, D,\)[/tex] and [tex]\(E\)[/tex] where [tex]\(A = B = 1\)[/tex] because the coefficients of [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] are 1. So,
[tex]\[ A = 1, \quad B = 1 \][/tex]
Comparing the coefficients of the expanded equation [tex]\(x^2 + y^2 - 2ky + (k^2 - 9) = 0\)[/tex] with [tex]\(A x^2 + B y^2 + C x + D y + E = 0\)[/tex], we find that:
[tex]\[ C = 0,\quad D = -2k,\quad E = k^2 - 9 \][/tex]
We can now check the given options to find a matching set of values. For each option:
A. [tex]\(A = 0, B = 0, C = 2, D = 2, E = 3\)[/tex]
- This cannot represent a circle since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] need to be non-zero.
B. [tex]\(A = 1, B = 1, C = 8, D = 0, E = 9\)[/tex]
- This matches the necessary forms and conditions given a center on the [tex]\(y\)[/tex]-axis.
C. [tex]\(A = 1, B = 1, C = 0, D = -8, E = 7\)[/tex]
- Here, [tex]\(D = -8\)[/tex] does not fit as [tex]\(k = 4\)[/tex] would suggest [tex]\(E = k^2 - 9 = 16 - 9 = 7,\)[/tex] but we need to match all conditions correctly.
D. [tex]\(A = 1, B = 1, C = -8, D = 0, E = 0\)[/tex]
- This does not fit our necessary form.
E. [tex]\(A = 1, B = 1, C = 8, D = 8, E = 3\)[/tex]
- This again, does not reduce correctly to our format, especially comparing the [tex]\(D\)[/tex] and [tex]\(E\)[/tex] values.
Thus, the set of values that correctly satisfies the equation of the circle is:
[tex]\[ A = 1, B = 1, C = 8, D = 0, E = 9 \][/tex]
Therefore, the correct answer is:
[tex]\[B.\ A = 1, B = 1, C = 8, D = 0, E = 9\][/tex]
The center of the circle [tex]\((h, k)\)[/tex] can be found by recognizing that it can be derived from the general form. The given condition states that the center lies on the [tex]\(y\)[/tex]-axis, meaning [tex]\(h = 0\)[/tex], and the radius [tex]\(R = 3\)[/tex].
The standard form of the circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = R^2 \][/tex]
Since [tex]\(h = 0\)[/tex]:
[tex]\[ x^2 + (y - k)^2 = 9 \][/tex]
Expanding and rearranging this into the general form:
[tex]\[ x^2 + y^2 - 2ky + k^2 = 9 \][/tex]
We need to identify [tex]\(A, B, C, D,\)[/tex] and [tex]\(E\)[/tex] where [tex]\(A = B = 1\)[/tex] because the coefficients of [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] are 1. So,
[tex]\[ A = 1, \quad B = 1 \][/tex]
Comparing the coefficients of the expanded equation [tex]\(x^2 + y^2 - 2ky + (k^2 - 9) = 0\)[/tex] with [tex]\(A x^2 + B y^2 + C x + D y + E = 0\)[/tex], we find that:
[tex]\[ C = 0,\quad D = -2k,\quad E = k^2 - 9 \][/tex]
We can now check the given options to find a matching set of values. For each option:
A. [tex]\(A = 0, B = 0, C = 2, D = 2, E = 3\)[/tex]
- This cannot represent a circle since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] need to be non-zero.
B. [tex]\(A = 1, B = 1, C = 8, D = 0, E = 9\)[/tex]
- This matches the necessary forms and conditions given a center on the [tex]\(y\)[/tex]-axis.
C. [tex]\(A = 1, B = 1, C = 0, D = -8, E = 7\)[/tex]
- Here, [tex]\(D = -8\)[/tex] does not fit as [tex]\(k = 4\)[/tex] would suggest [tex]\(E = k^2 - 9 = 16 - 9 = 7,\)[/tex] but we need to match all conditions correctly.
D. [tex]\(A = 1, B = 1, C = -8, D = 0, E = 0\)[/tex]
- This does not fit our necessary form.
E. [tex]\(A = 1, B = 1, C = 8, D = 8, E = 3\)[/tex]
- This again, does not reduce correctly to our format, especially comparing the [tex]\(D\)[/tex] and [tex]\(E\)[/tex] values.
Thus, the set of values that correctly satisfies the equation of the circle is:
[tex]\[ A = 1, B = 1, C = 8, D = 0, E = 9 \][/tex]
Therefore, the correct answer is:
[tex]\[B.\ A = 1, B = 1, C = 8, D = 0, E = 9\][/tex]
