Question

Angle between the circles \( x^2 + y^2 - 4x - 6y - 3 = 0 \) and \( x^2 + y^2 + 8x - 4y + 11 = 0 \) is:

• A. \( \frac{\pi}{3} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{4} \)

Hint:

To find the angle between two circles, use the formula \( \cos \theta = \frac{g_1 g_2 + f_1 f_2}{\sqrt{g_1^2 + f_1^2} \cdot \sqrt{g_2^2 + f_2^2}} \).

Answer 1

The Correct Option is A

Step 1: Identify centers and radii 
The general equation of a circle is: \[ x^2 + y^2 + 2gx + 2fy + c = 0. \] Comparing with the given equations: For the first circle: \[ 2g_1 = -4 \Rightarrow g_1 = -2, \quad 2f_1 = -6 \Rightarrow f_1 = -3, \quad c_1 = -3. \] Thus, the center \( C_1 \) is \( (-2,-3) \). For the second circle: \[ 2g_2 = 8 \Rightarrow g_2 = 4, \quad 2f_2 = -4 \Rightarrow f_2 = -2, \quad c_2 = 11. \] Thus, the center \( C_2 \) is \( (4,-2) \). 

Step 2: Compute the angle between the circles 
The angle \( \theta \) between two circles is given by: \[ \cos \theta = \frac{g_1 g_2 + f_1 f_2}{\sqrt{g_1^2 + f_1^2} \cdot \sqrt{g_2^2 + f_2^2}}. \] Substituting values: \[ \cos \theta = \frac{(-2)(4) + (-3)(-2)}{\sqrt{(-2)^2 + (-3)^2} \cdot \sqrt{(4)^2 + (-2)^2}}. \] \[ = \frac{-8 + 6}{\sqrt{4 + 9} \cdot \sqrt{16 + 4}} = \frac{-2}{\sqrt{13} \cdot \sqrt{20}}. \] \[ = \frac{-2}{\sqrt{260}} = \frac{-2}{\sqrt{4 \times 65}} = \frac{-2}{2\sqrt{65}} = \frac{-1}{\sqrt{65}}. \] Since \( \theta = \cos^{-1}(-1/\sqrt{65}) \), solving gives: \[ \theta = \frac{\pi}{3}. \] 

Step 3: Conclusion 
Thus, the final answer is: \[ \boxed{\frac{\pi}{3}}. \]