Question

If the line \( x \cos \alpha + y \sin \alpha = p \) represents the common chord of the circles \( x^2 + y^2 = a^2 \) and \( x^2 + y^2 + b^2 = 2b \), where \( a > b \), where A and B lie on the first circle and P and Q lie on the second circle, then \( AP \) is equal to:

• A. \( \sqrt{a^2 + p^2} + \sqrt{b^2 + p^2} \)
• B. \( \sqrt{a^2 - p^2} + \sqrt{b^2 - p^2} \)
• C. \( \sqrt{a^2 + p^2} - \sqrt{b^2 + p^2} \)
• D. \( \sqrt{a^2 - p^2} - \sqrt{b^2 - p^2} \)

Hint:

To find the length of a common chord, use geometric properties of the intersecting circles.

Answer 1

The Correct Option is B

Step 1: Solve using geometry of circles.
By applying the properties of the common chord of two circles, we use the formula for the length of the chord to find that \( AP = \sqrt{a^2 - p^2} + \sqrt{b^2 - p^2} \).
Thus, the correct answer is (2).