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

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 can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the given circle equation in standard form. The equation is:
\[ x^2 + y^2 + 4x - 6y - 12 = 0 \]
We can complete the square for \(x\) and \(y\).

For \(x\):
\[ x^2 + 4x = (x + 2)^2 - 4 \]

For \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

Substituting these back into the equation:
\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 - 12 = 0 \]
\[ (x + 2)^2 + (y - 3)^2 - 25 = 0 \]
\[ (x + 2)^2 + (y - 3)^2 = 25 \]

### Step 2: Identify the Center and Radius
From the standard form \((x + 2)^2 + (y - 3)^2 = 25\), we can identify:
- Center \(C(-2, 3)\)
- Radius \(r = 5\)

### Step 3: Understand the Geometry of the Problem
The problem states that the angle subtended by the chord at the circumference is \(\frac{\pi}{3}\) radians (or \(60^\circ\)). By the properties of circles, the angle at the center corresponding to this angle at the circumference will be:
\[ \text{Angle at center} = 2 \times \text{Angle at circumference} = 2 \times \frac{\pi}{3} = \frac{2\pi}{3} \]

### Step 4: Use the Sine Rule
Let \(P(h, k)\) be the midpoint of the chord \(AB\). The distance from the center \(C\) to the midpoint \(P\) can be found using the sine of the angle at the center:
\[ CP = r \cdot \sin\left(\frac{\text{Angle at center}}{2}\right) = 5 \cdot \sin\left(\frac{2\pi/3}{2}\right) = 5 \cdot \sin\left(\frac{\pi}{3}\right) = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \]

### Step 5: Set Up the Distance Equation
The distance \(CP\) can also be expressed using the distance formula:
\[ CP = \sqrt{(h + 2)^2 + (k - 3)^2} \]
Setting this equal to \(\frac{5\sqrt{3}}{2}\):
\[ \sqrt{(h + 2)^2 + (k - 3)^2} = \frac{5\sqrt{3}}{2} \]

### Step 6: Square Both Sides
Squaring both sides gives:
\[ (h + 2)^2 + (k - 3)^2 = \left(\frac{5\sqrt{3}}{2}\right)^2 \]
\[ (h + 2)^2 + (k - 3)^2 = \frac{75}{4} \]

### Step 7: Replace \(h\) and \(k\) with \(x\) and \(y\)
Replacing \(h\) with \(x\) and \(k\) with \(y\):
\[ (x + 2)^2 + (y - 3)^2 = \frac{75}{4} \]

### Step 8: Final Equation
This represents the locus of the midpoints of the chords that subtend an angle of \(\frac{\pi}{3}\) radians at the circumference:
\[ (x + 2)^2 + (y - 3)^2 = 18.75 \]