Question

The angle of intersection of the circles \( x^2 + y^2 - 8x - 9 = 0 \) and \( x^2 + y^2 + 2x - 4y - 11 = 0 \) is

• A. \( \tan^{-1} \left( \frac{9}{8} \right) \)
• B. \( \tan^{-1} \left( 19 \right) \)
• C. \( \tan^{-1} \left( 5 \right) \)
• D. \( \tan^{-1} \left( 1 \right) \)

Hint:

The angle of intersection of two curves can be calculated using the formula \( \theta = \tan^{-1} \left( \frac{m_1 - m_2}{1 + m_1 m_2} \right) \), where \( m_1 \) and \( m_2 \) are the slopes of the tangents.

Answer 1

The Correct Option is A

Step 1: Analyzing the intersection.
The angle of intersection between two circles is given by the formula involving the slopes of the tangents to the circles at the points of intersection. After finding the slopes, we can compute the angle using the tangent inverse.

Step 2: Conclusion.
The angle of intersection is \( \tan^{-1} \left( \frac{9}{8} \right) \), corresponding to option (1).