The equation of the circle concentric with `x^(2)+y^(2)-2x+8y-23=0` and passing through (2, 3) is

Options

A. `x^(2)+y^(2)-6x+4y-12=0`
B. `x^(2)+y^(2)-2x+8y-33=0`
C. `x^(2)+y^(2)+6x-4y-12=0`
D. `x^(2)+y^(2)+x+8y+33=0`

Answer

To find the equation of the circle that is concentric with the given circle \(x^2 + y^2 - 2x + 8y - 23 = 0\) and passes through the point (2, 3), we will follow these steps:

### Step 1: Convert the given circle equation to standard form
The given equation is:
\[ x^2 + y^2 - 2x + 8y - 23 = 0 \]
Rearranging it, we have:
\[ x^2 - 2x + y^2 + 8y = 23 \]

### Step 2: Complete the square for \(x\) and \(y\)
For \(x\):
- Take the coefficient of \(x\) which is \(-2\), halve it to get \(-1\), and square it to get \(1\).
- Add and subtract \(1\).

For \(y\):
- Take the coefficient of \(y\) which is \(8\), halve it to get \(4\), and square it to get \(16\).
- Add and subtract \(16\).

Now we can rewrite the equation:
\[ (x^2 - 2x + 1) + (y^2 + 8y + 16) = 23 + 1 + 16 \]
This simplifies to:
\[ (x - 1)^2 + (y + 4)^2 = 40 \]

### Step 3: Identify the center and radius of the given circle
From the standard form \((x - 1)^2 + (y + 4)^2 = 40\), we can see:
- The center of the circle is \((1, -4)\).
- The radius \(r\) is \(\sqrt{40}\).

### Step 4: Write the equation of the concentric circle
Since the new circle is concentric with the given circle, it will have the same center \((1, -4)\) but a different radius \(r_1\). The equation of the new circle is:
\[ (x - 1)^2 + (y + 4)^2 = r_1^2 \]

### Step 5: Find the radius using the point (2, 3)
Since the new circle passes through the point (2, 3), we substitute \(x = 2\) and \(y = 3\) into the equation:
\[ (2 - 1)^2 + (3 + 4)^2 = r_1^2 \]
Calculating this gives:
\[ 1^2 + 7^2 = r_1^2 \]
\[ 1 + 49 = r_1^2 \]
\[ r_1^2 = 50 \]

### Step 6: Write the final equation of the new circle
Now we have:
\[ (x - 1)^2 + (y + 4)^2 = 50 \]

### Step 7: Convert to general form
Expanding the equation:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ (y + 4)^2 = y^2 + 8y + 16 \]
Combining these gives:
\[ x^2 - 2x + 1 + y^2 + 8y + 16 = 50 \]
Simplifying:
\[ x^2 + y^2 - 2x + 8y + 17 - 50 = 0 \]
\[ x^2 + y^2 - 2x + 8y - 33 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 2x + 8y - 33 = 0 \]

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