The locus of the mid points of the chords of the circle `x^2+y^2+4x-6y-12=0` which subtend an angle of `pi/3`radians at its circumference is:
(A) `(x-2)^2+(y+3)^2=6.25` (B) `(x+2)^2+(y-3)^2=6.25`
(C) `(x+2)^2+(y-3)^2=18.75` (D) `(x+2)^2+(y+3)^2=18.75`

Options

A. `(x+2)^(2)+(y+3)^(2)=6.25`
B. `(x-2)^(2)+(y+3)^(2)=6.25`
C. `(x+2)^(2)+(y-3)^(2)=18.75`
D. `(x+2)^(2)+(y+3)^(2)=18.75`

Answer

To find the locus of the midpoints of the chords of the circle \(x^2 + y^2 + 4x - 6y - 12 = 0\) that subtend an angle of \(\frac{\pi}{3}\) radians at the circumference, we will follow these steps:

### Step 1: Rewrite the equation of the circle in standard form
The given equation is:
\[ x^2 + y^2 + 4x - 6y - 12 = 0 \]
We can rearrange this to complete the square for \(x\) and \(y\).

1. Group the \(x\) and \(y\) terms:
\[ (x^2 + 4x) + (y^2 - 6y) = 12 \]

2. Complete the square:
- For \(x^2 + 4x\), add and subtract \(4\):
\[ (x + 2)^2 - 4 \]
- For \(y^2 - 6y\), add and subtract \(9\):
\[ (y - 3)^2 - 9 \]

3. Substitute back into the equation:
\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 12 \]
Simplifying gives:
\[ (x + 2)^2 + (y - 3)^2 = 25 \]

Thus, the center of the circle is \((-2, 3)\) and the radius is \(5\).

### Step 2: Understanding the geometry of the problem
Let \(O\) be the center of the circle at \((-2, 3)\) and let \(H(h, k)\) be the midpoint of the chord \(AB\) that subtends an angle of \(\frac{\pi}{3}\) at the circumference.

### Step 3: Using the angle subtended at the center
The angle subtended at the center \(O\) is twice that at the circumference. Therefore, the angle \(\angle AOB = 2 \cdot \frac{\pi}{3} = \frac{2\pi}{3}\).

### Step 4: Relating the midpoint to the radius
Using the cosine rule in triangle \(OAP\) (where \(P\) is a point on the circle):
\[ \cos\left(\frac{\pi}{3}\right) = \frac{OP}{R} \]
Where \(R\) is the radius of the circle (which is \(5\)):
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Thus:
\[ \frac{OP}{5} = \frac{1}{2} \implies OP = \frac{5}{2} \]

### Step 5: Setting up the distance formula
The distance \(OP\) can also be expressed using the coordinates of the midpoint \(H(h, k)\):
\[ OP = \sqrt{(h + 2)^2 + (k - 3)^2} \]
Setting this equal to \(\frac{5}{2}\):
\[ \sqrt{(h + 2)^2 + (k - 3)^2} = \frac{5}{2} \]

### Step 6: Squaring both sides
Squaring both sides gives:
\[ (h + 2)^2 + (k - 3)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]

### Step 7: Replacing \(h\) and \(k\) with \(x\) and \(y\)
Since \(h\) and \(k\) represent the coordinates of the midpoint, we replace them with \(x\) and \(y\):
\[ (x + 2)^2 + (y - 3)^2 = \frac{25}{4} \]

### Step 8: Final form of the locus
This can be rewritten as:
\[ (x + 2)^2 + (y - 3)^2 = 6.25 \]

### Conclusion
Thus, the locus of the midpoints of the chords of the circle that subtend an angle of \(\frac{\pi}{3}\) radians at the circumference is:
\[ \boxed{(x + 2)^2 + (y - 3)^2 = 6.25} \]
This corresponds to option (B).