If (3,-2) is the midpoint of the chord AB of the circle `x^(2)+y^(2)-4x+6y-5=0" then AB="`

Options

A. 4
B. 8
C. 12
D. 16

Answer

To find the length of the chord AB of the circle given that (3, -2) is the midpoint, we can follow these steps:

### Step 1: Write the equation of the circle
The equation of the circle is given as:
\[ x^2 + y^2 - 4x + 6y - 5 = 0 \]

### Step 2: Identify the center and radius of the circle
To find the center and radius, we can rewrite the equation in standard form. We will complete the square for both x and y.

1. Rearranging the equation:
\[ (x^2 - 4x) + (y^2 + 6y) = 5 \]

2. Completing the square:
- For \(x^2 - 4x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
- For \(y^2 + 6y\):
\[ y^2 + 6y = (y + 3)^2 - 9 \]

3. Substitute back into the equation:
\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 = 5 \]
\[ (x - 2)^2 + (y + 3)^2 = 18 \]

Thus, the center of the circle is (2, -3) and the radius is \( \sqrt{18} = 3\sqrt{2} \).

### Step 3: Use the midpoint to find the length of the chord
The length of the chord can be calculated using the formula:
\[ \text{Length of chord} = 2 \sqrt{r^2 - d^2} \]
where \(r\) is the radius and \(d\) is the distance from the center of the circle to the midpoint of the chord.

### Step 4: Calculate the distance \(d\)
The distance \(d\) from the center (2, -3) to the midpoint (3, -2) is given by:
\[ d = \sqrt{(3 - 2)^2 + (-2 + 3)^2} \]
\[ d = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 5: Calculate the length of the chord
Now we can substitute the values into the chord length formula:
1. Radius \(r = 3\sqrt{2}\)
2. Distance \(d = \sqrt{2}\)

Substituting these into the formula:
\[ \text{Length of chord} = 2 \sqrt{(3\sqrt{2})^2 - (\sqrt{2})^2} \]
\[ = 2 \sqrt{18 - 2} \]
\[ = 2 \sqrt{16} \]
\[ = 2 \times 4 = 8 \]

### Conclusion
The length of the chord AB is \(8\).

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