Question

The radical axis of the circles \[ x^2 + y^2 + 2gx + 2fy + c = 0 \] and \[ 2x^2 + 2y^2 + 3x + 8y + 2c = 0 \] touches the circle \[ x^2 + y^2 + 2x + 2y + 1 = 0. \] Then:

• A. \( g = \frac{3}{8} \) \textbf{or} \( f = 1 \)
• B. \( g = \frac{2}{3} \) \textbf{or} \( f = 3 \)
• C. \( g = \frac{1}{2} \) \textbf{or} \( f = 1 \)
• D. \( g = \frac{3}{4} \) \textbf{or} \( f = 2 \)

Hint:

For radical axis problems, subtract the given circle equations and use the touching condition with the third circle to solve for unknowns.

Answer 1

The Correct Option is D

Step 1: Finding the radical axis equation The radical axis is given by subtracting the two circle equations: \[ (x^2 + y^2 + 2gx + 2fy + c) - (2x^2 + 2y^2 + 3x + 8y + 2c) = 0. \] Simplifying, \[ -x^2 - y^2 + (2g - 3)x + (2f - 8)y + c - 2c = 0. \] Step 2: Applying the touching condition For the radical axis to touch the third circle, its perpendicular distance from the center must equal its radius. Solving for \( g \) and \( f \), \[ g = \frac{3}{4}, \quad f = 2. \]