Question

Let P and Q be the inverse points with respect to the circle \( S = x^2 + y^2 - 4x - 6y + k = 0 \), and C be the center of the circle. If \( CP.CQ = 4 \), and \( P = (1, 2) \), then \( Q = (a, b) \) and \( 2a = \dots \)

• A. \( b \)
• B. \( -1 \)
• C. \( 3b \)
• D. \( 0 \)

Hint:

For inverse points with respect to a circle, use the relationship \( CP \times CQ = r^2 \), where \( r \) is the radius of the circle.

Answer 1

The Correct Option is B

The equation of the circle is: \[ S = x^2 + y^2 - 4x - 6y + k = 0 \] The center \( C \) of the circle is \( (2, 3) \), and the distance \( CP \times CQ = 4 \). Using the inverse point relationship and the given data, we find that: \[ 2a = -1 \] % Final Answer The value of \( 2a \) is \( -1 \).