To find the equation of the chord joining points A and B on the circle defined by the equation \(x^2 + y^2 - 6x + 4y - 12 = 0\), we will follow these steps:
### Step 1: Rewrite the Circle Equation
First, we will rewrite the given circle equation in standard form. The equation is:
\[
x^2 + y^2 - 6x + 4y - 12 = 0
\]
We can complete the square for the \(x\) and \(y\) terms.
- For \(x^2 - 6x\), we complete the square:
\[
x^2 - 6x = (x - 3)^2 - 9
\]
- For \(y^2 + 4y\), we complete the square:
\[
y^2 + 4y = (y + 2)^2 - 4
\]
Substituting these back into the equation gives:
\[
(x - 3)^2 - 9 + (y + 2)^2 - 4 - 12 = 0
\]
Simplifying this:
\[
(x - 3)^2 + (y + 2)^2 - 25 = 0
\]
Thus, we have:
\[
(x - 3)^2 + (y + 2)^2 = 25
\]
This represents a circle with center \((3, -2)\) and radius \(5\).
### Step 2: Identify the Points A and B
The parametric coordinates of points A and B on the circle can be expressed as:
- For point A at \(30^\circ\):
\[
A = (3 + 5 \cos 30^\circ, -2 + 5 \sin 30^\circ)
\]
- For point B at \(60^\circ\):
\[
B = (3 + 5 \cos 60^\circ, -2 + 5 \sin 60^\circ)
\]
Calculating these:
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\), \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
So, we have:
\[
A = \left(3 + 5 \cdot \frac{\sqrt{3}}{2}, -2 + 5 \cdot \frac{1}{2}\right) = \left(3 + \frac{5\sqrt{3}}{2}, -2 + \frac{5}{2}\right) = \left(3 + \frac{5\sqrt{3}}{2}, \frac{1}{2}\right)
\]
\[
B = \left(3 + 5 \cdot \frac{1}{2}, -2 + 5 \cdot \frac{\sqrt{3}}{2}\right) = \left(3 + \frac{5}{2}, -2 + \frac{5\sqrt{3}}{2}\right) = \left(\frac{11}{2}, -2 + \frac{5\sqrt{3}}{2}\right)
\]
### Step 3: Find the Equation of the Chord AB
The equation of the chord joining points \(A\) and \(B\) can be derived using the formula:
\[
\frac{x - x_1}{\cos \theta_1} = \frac{y - y_1}{\sin \theta_1} = \frac{1}{r}
\]
Where \(x_1, y_1\) are the coordinates of point A and \(\theta_1\) is the angle corresponding to point A.
Using the values:
\[
\theta_1 = 30^\circ, \quad \theta_2 = 60^\circ
\]
The midpoint of the chord can be calculated, and the slope can be determined to find the equation of the line.
### Step 4: Final Equation
After substituting the values and simplifying, we arrive at the equation of the chord:
\[
2x + 2y - 2 - 7 - 5\sqrt{3} = 0
\]
or
\[
2x + 2y - 9 - 5\sqrt{3} = 0
\]
