Answer :
To determine the general form of the equation of the circle with its center at \((-2, 1)\) and passing through \((-4, 1)\), we can follow these steps:
1. Identify the radius of the circle:
- The center of the circle is \((-2, 1)\).
- A point on the circle is \((-4, 1)\).
- The radius \(r\) can be found using the distance formula:
[tex]\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where \((x_1, y_1)\) is the center and \((x_2, y_2)\) is the point on the circle.
Substituting in our values:
[tex]\[ r = \sqrt{(-4 - (-2))^2 + (1 - 1)^2} = \sqrt{(-4 + 2)^2 + 0^2} = \sqrt{(-2)^2} = \sqrt{4} = 2 \][/tex]
2. Set up the standard form of the circle's equation:
- The standard form of the equation for a circle with center \((h, k)\) and radius \(r\) is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
- For this specific circle, \(h = -2\), \(k = 1\), and \(r = 2\):
[tex]\[ (x + 2)^2 + (y - 1)^2 = 4 \][/tex]
3. Expand and simplify the equation:
- Expanding \((x + 2)^2\):
[tex]\[ (x + 2)^2 = x^2 + 4x + 4 \][/tex]
- Expanding \((y - 1)^2\):
[tex]\[ (y - 1)^2 = y^2 - 2y + 1 \][/tex]
- Putting it all together:
[tex]\[ x^2 + 4x + 4 + y^2 - 2y + 1 = 4 \][/tex]
- Simplifying the equation:
[tex]\[ x^2 + y^2 + 4x - 2y + 5 = 4 \][/tex]
[tex]\[ x^2 + y^2 + 4x - 2y + 1 = 0 \][/tex]
4. Match with the given options:
- The simplified equation \(x^2 + y^2 + 4x - 2y + 1 = 0\) corresponds to option B.
Therefore, the correct answer is:
[tex]\[ \boxed{x^2 + y^2 + 4x - 2y + 1 = 0} \][/tex]
1. Identify the radius of the circle:
- The center of the circle is \((-2, 1)\).
- A point on the circle is \((-4, 1)\).
- The radius \(r\) can be found using the distance formula:
[tex]\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where \((x_1, y_1)\) is the center and \((x_2, y_2)\) is the point on the circle.
Substituting in our values:
[tex]\[ r = \sqrt{(-4 - (-2))^2 + (1 - 1)^2} = \sqrt{(-4 + 2)^2 + 0^2} = \sqrt{(-2)^2} = \sqrt{4} = 2 \][/tex]
2. Set up the standard form of the circle's equation:
- The standard form of the equation for a circle with center \((h, k)\) and radius \(r\) is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
- For this specific circle, \(h = -2\), \(k = 1\), and \(r = 2\):
[tex]\[ (x + 2)^2 + (y - 1)^2 = 4 \][/tex]
3. Expand and simplify the equation:
- Expanding \((x + 2)^2\):
[tex]\[ (x + 2)^2 = x^2 + 4x + 4 \][/tex]
- Expanding \((y - 1)^2\):
[tex]\[ (y - 1)^2 = y^2 - 2y + 1 \][/tex]
- Putting it all together:
[tex]\[ x^2 + 4x + 4 + y^2 - 2y + 1 = 4 \][/tex]
- Simplifying the equation:
[tex]\[ x^2 + y^2 + 4x - 2y + 5 = 4 \][/tex]
[tex]\[ x^2 + y^2 + 4x - 2y + 1 = 0 \][/tex]
4. Match with the given options:
- The simplified equation \(x^2 + y^2 + 4x - 2y + 1 = 0\) corresponds to option B.
Therefore, the correct answer is:
[tex]\[ \boxed{x^2 + y^2 + 4x - 2y + 1 = 0} \][/tex]
