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To find the equation of the circle concentric with the circle [tex]\(x^2 + y^2 + 4x + 6y + 11 = 0\)[/tex] and passing through the point [tex]\(A(5,4)\)[/tex], follow these steps:
1. Find the center of the original circle:
The given equation of the circle is in the general form:
[tex]\[ x^2 + y^2 + 4x + 6y + 11 = 0 \][/tex]
To convert this to the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 4x + y^2 + 6y + 11 = 0 \][/tex]
Completing the square:
[tex]\[ (x^2 + 4x) + (y^2 + 6y) + 11 = 0 \][/tex]
For [tex]\(x\)[/tex]:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
For [tex]\(y\)[/tex]:
[tex]\[ y^2 + 6y = (y + 3)^2 - 9 \][/tex]
Substitute these back into the equation:
[tex]\[ (x + 2)^2 - 4 + (y + 3)^2 - 9 + 11 = 0 \][/tex]
Simplify:
[tex]\[ (x + 2)^2 + (y + 3)^2 - 2 = 0 \][/tex]
[tex]\[ (x + 2)^2 + (y + 3)^2 = 2 \][/tex]
The center of the circle is [tex]\((-2, -3)\)[/tex].
2. Determine the radius of the new circle:
Since the new circle is concentric with the original, it has the same center [tex]\((-2, -3)\)[/tex]. To find the radius, we calculate the distance between the center and the given point [tex]\(A(5, 4)\)[/tex]:
[tex]\[ \text{radius} = \sqrt{(5 - (-2))^2 + (4 - (-3))^2} \][/tex]
[tex]\[ = \sqrt{(5 + 2)^2 + (4 + 3)^2} \][/tex]
[tex]\[ = \sqrt{7^2 + 7^2} \][/tex]
[tex]\[ = \sqrt{49 + 49} \][/tex]
[tex]\[ = \sqrt{98} \][/tex]
[tex]\[ = 7\sqrt{2} \][/tex]
3. Form the equation of the new circle:
The standard form of the new circle with radius [tex]\(7\sqrt{2}\)[/tex] is:
[tex]\[ (x + 2)^2 + (y + 3)^2 = (7\sqrt{2})^2 \][/tex]
Simplify the right side:
[tex]\[ (x + 2)^2 + (y + 3)^2 = 98 \][/tex]
4. Convert the new circle's equation to the general form:
Expanding the left side:
[tex]\[ (x + 2)^2 = x^2 + 4x + 4 \][/tex]
[tex]\[ (y + 3)^2 = y^2 + 6y + 9 \][/tex]
Substitute back into the equation:
[tex]\[ x^2 + 4x + 4 + y^2 + 6y + 9 = 98 \][/tex]
Combine and simplify terms:
[tex]\[ x^2 + y^2 + 4x + 6y + 13 = 98 \][/tex]
[tex]\[ x^2 + y^2 + 4x + 6y + 13 - 98 = 0 \][/tex]
[tex]\[ x^2 + y^2 + 4x + 6y - 85 = 0 \][/tex]
Therefore, the equation of the new circle is:
[tex]\[ x^2 + y^2 + 4x + 6y - 85 = 0 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{a. } x^2 + y^2 + 4x + 6y - 85 = 0} \][/tex]
1. Find the center of the original circle:
The given equation of the circle is in the general form:
[tex]\[ x^2 + y^2 + 4x + 6y + 11 = 0 \][/tex]
To convert this to the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 4x + y^2 + 6y + 11 = 0 \][/tex]
Completing the square:
[tex]\[ (x^2 + 4x) + (y^2 + 6y) + 11 = 0 \][/tex]
For [tex]\(x\)[/tex]:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
For [tex]\(y\)[/tex]:
[tex]\[ y^2 + 6y = (y + 3)^2 - 9 \][/tex]
Substitute these back into the equation:
[tex]\[ (x + 2)^2 - 4 + (y + 3)^2 - 9 + 11 = 0 \][/tex]
Simplify:
[tex]\[ (x + 2)^2 + (y + 3)^2 - 2 = 0 \][/tex]
[tex]\[ (x + 2)^2 + (y + 3)^2 = 2 \][/tex]
The center of the circle is [tex]\((-2, -3)\)[/tex].
2. Determine the radius of the new circle:
Since the new circle is concentric with the original, it has the same center [tex]\((-2, -3)\)[/tex]. To find the radius, we calculate the distance between the center and the given point [tex]\(A(5, 4)\)[/tex]:
[tex]\[ \text{radius} = \sqrt{(5 - (-2))^2 + (4 - (-3))^2} \][/tex]
[tex]\[ = \sqrt{(5 + 2)^2 + (4 + 3)^2} \][/tex]
[tex]\[ = \sqrt{7^2 + 7^2} \][/tex]
[tex]\[ = \sqrt{49 + 49} \][/tex]
[tex]\[ = \sqrt{98} \][/tex]
[tex]\[ = 7\sqrt{2} \][/tex]
3. Form the equation of the new circle:
The standard form of the new circle with radius [tex]\(7\sqrt{2}\)[/tex] is:
[tex]\[ (x + 2)^2 + (y + 3)^2 = (7\sqrt{2})^2 \][/tex]
Simplify the right side:
[tex]\[ (x + 2)^2 + (y + 3)^2 = 98 \][/tex]
4. Convert the new circle's equation to the general form:
Expanding the left side:
[tex]\[ (x + 2)^2 = x^2 + 4x + 4 \][/tex]
[tex]\[ (y + 3)^2 = y^2 + 6y + 9 \][/tex]
Substitute back into the equation:
[tex]\[ x^2 + 4x + 4 + y^2 + 6y + 9 = 98 \][/tex]
Combine and simplify terms:
[tex]\[ x^2 + y^2 + 4x + 6y + 13 = 98 \][/tex]
[tex]\[ x^2 + y^2 + 4x + 6y + 13 - 98 = 0 \][/tex]
[tex]\[ x^2 + y^2 + 4x + 6y - 85 = 0 \][/tex]
Therefore, the equation of the new circle is:
[tex]\[ x^2 + y^2 + 4x + 6y - 85 = 0 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{a. } x^2 + y^2 + 4x + 6y - 85 = 0} \][/tex]
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