Question

If the power of the point \( (1,6) \) with respect to the circle \( x^2+y^2+4x-6y-a=0 \) is \( -16 \), then \( a \) is:

• A. \( 7 \)
• B. \( 11 \)
• C. \( 13 \)
• D. \( 21 \)

Hint:

The power of a point with respect to a circle helps determine its relative position: - \( P>0 \) → Point is outside the circle - \( P = 0 \) → Point lies on the circle - \( P<0 \) → Point is inside the circle

Answer 1

The Correct Option is C

The power of a point \( (x_0, y_0) \) with respect to a circle \( x^2+y^2+Dx+Ey+F=0 \) is given by: \[ P = x_0^2 + y_0^2 + D x_0 + E y_0 + F \] For the given circle: \[ D = 4, E = -6, F = -a \] Substituting \( (x_0, y_0) = (1,6) \): \[ P = (1)^2 + (6)^2 + (4)(1) + (-6)(6) - a \] \[ = 1 + 36 + 4 - 36 - a \] \[ = 5 - a \] Given that \( P = -16 \): \[ 5 - a = -16 \] \[ a = 13 \] Thus, the correct answer is \( 13 \).