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Sagot :
The problem requires us to find the correct set of values for [tex]\( A, B, C, D \)[/tex], and [tex]\( E \)[/tex] that describe the equation of a circle with a radius of 3 units and whose center lies on the [tex]\( y \)[/tex]-axis.
First, let's recall the general form of a circle's equation in Cartesian coordinates. When the center of the circle is at [tex]\((h, k)\)[/tex] and the radius is [tex]\(r\)[/tex], the equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Given:
- The radius [tex]\( r = 3 \)[/tex].
- The center lies on the [tex]\( y \)[/tex]-axis, meaning [tex]\( h = 0 \)[/tex].
Therefore, the equation of the circle becomes:
[tex]\[ x^2 + (y - k)^2 = 3^2 \quad \text{or} \quad x^2 + (y - k)^2 = 9 \][/tex]
Expanding [tex]\((y - k)^2\)[/tex]:
[tex]\[ (y - k)^2 = y^2 - 2ky + k^2 \][/tex]
Substituting this back into the circle's equation, we get:
[tex]\[ x^2 + y^2 - 2ky + k^2 = 9 \][/tex]
Now, we want to fit this equation into the general form:
[tex]\[ A x^2 + B y^2 + C x + D y + E = 0 \][/tex]
Given that [tex]\( A = B \)[/tex] and both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] must not be zero, we set [tex]\( A = 1 \)[/tex] and [tex]\( B = 1 \)[/tex]:
[tex]\[ 1 \cdot x^2 + 1 \cdot y^2 + 0 \cdot x + D y + E = 0 \quad \text{(Comparing coefficients, we get the equations for} \, C, D, \, \text{and} \, E)}. \][/tex]
Identifying from the expanded circle equation:
[tex]\[ x^2 + y^2 - 2ky + k^2 = 9 \][/tex]
Compare both forms, we get:
[tex]\[ D = -2k \quad \text{and} \quad E = k^2 - 9 \][/tex]
By inspection, considering our values comparing it to possible sets of [tex]\( A, B, C, D, \)[/tex] and [tex]\( E \)[/tex], we see the correct option satisfying the circle's requirement fits as follows:
- [tex]\( A = 1 \)[/tex]
- [tex]\( B = 1 \)[/tex]
- There should be no [tex]\( x \)[/tex] term, therefore [tex]\( C = 0 \)[/tex]
- Preliminarily can check all answers and see the one fitting which central on y-axis calculation and maintaining given circle equation.
So, looking at option B provides the [tex]\( C \)[/tex] aligned with simplifying and one containing correct values:
[tex]\[ A=1, B=1, C=8, D=0, E=9. \][/tex]
Thus, the correct set of values for [tex]\( A, B, C, D, \)[/tex] and [tex]\( E \)[/tex] that correspond to the equation of the circle is:
[tex]\[ \boxed{B: A=1, B=1, C=8, D=0, E=9} \][/tex]
First, let's recall the general form of a circle's equation in Cartesian coordinates. When the center of the circle is at [tex]\((h, k)\)[/tex] and the radius is [tex]\(r\)[/tex], the equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Given:
- The radius [tex]\( r = 3 \)[/tex].
- The center lies on the [tex]\( y \)[/tex]-axis, meaning [tex]\( h = 0 \)[/tex].
Therefore, the equation of the circle becomes:
[tex]\[ x^2 + (y - k)^2 = 3^2 \quad \text{or} \quad x^2 + (y - k)^2 = 9 \][/tex]
Expanding [tex]\((y - k)^2\)[/tex]:
[tex]\[ (y - k)^2 = y^2 - 2ky + k^2 \][/tex]
Substituting this back into the circle's equation, we get:
[tex]\[ x^2 + y^2 - 2ky + k^2 = 9 \][/tex]
Now, we want to fit this equation into the general form:
[tex]\[ A x^2 + B y^2 + C x + D y + E = 0 \][/tex]
Given that [tex]\( A = B \)[/tex] and both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] must not be zero, we set [tex]\( A = 1 \)[/tex] and [tex]\( B = 1 \)[/tex]:
[tex]\[ 1 \cdot x^2 + 1 \cdot y^2 + 0 \cdot x + D y + E = 0 \quad \text{(Comparing coefficients, we get the equations for} \, C, D, \, \text{and} \, E)}. \][/tex]
Identifying from the expanded circle equation:
[tex]\[ x^2 + y^2 - 2ky + k^2 = 9 \][/tex]
Compare both forms, we get:
[tex]\[ D = -2k \quad \text{and} \quad E = k^2 - 9 \][/tex]
By inspection, considering our values comparing it to possible sets of [tex]\( A, B, C, D, \)[/tex] and [tex]\( E \)[/tex], we see the correct option satisfying the circle's requirement fits as follows:
- [tex]\( A = 1 \)[/tex]
- [tex]\( B = 1 \)[/tex]
- There should be no [tex]\( x \)[/tex] term, therefore [tex]\( C = 0 \)[/tex]
- Preliminarily can check all answers and see the one fitting which central on y-axis calculation and maintaining given circle equation.
So, looking at option B provides the [tex]\( C \)[/tex] aligned with simplifying and one containing correct values:
[tex]\[ A=1, B=1, C=8, D=0, E=9. \][/tex]
Thus, the correct set of values for [tex]\( A, B, C, D, \)[/tex] and [tex]\( E \)[/tex] that correspond to the equation of the circle is:
[tex]\[ \boxed{B: A=1, B=1, C=8, D=0, E=9} \][/tex]
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