Question

If the circle passing through the points \( (3,5), (5,5), (3,-3) \) cuts the circle \( x^2 + y^2 + 2x + 2fy = 0 \) orthogonally, then the value of \( f \) is:

• A. \(-12\)
• B. \(-3\)
• C. \(-15\)
• D. \(-4\)

Hint:

If two circles intersect orthogonally, then \( 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \), where the circles are in general form: \( x^2 + y^2 + 2gx + 2fy + c = 0 \)

Answer 1

The Correct Option is D

Step 1: General equation of circle passing through three points 
Let the required circle be: \[ x^2 + y^2 + Dx + Ey + F = 0 \] 

Step 2: Solve the system of equations (1), (2), (3) 
Subtract (1) from (2): \[ (5D + 5E + F) - (3D + 5E + F) = -50 + 34 \Rightarrow 2D = -16 \Rightarrow D = -8 \] Subtract (1) from (3): \[ (3D - 3E + F) - (3D + 5E + F) = -18 + 34 \Rightarrow -8E = 16 \Rightarrow E = -2 \] Now substitute \( D = -8, E = -2 \) into (1): \[ 3(-8) + 5(-2) + F = -34 \Rightarrow -24 -10 + F = -34 \Rightarrow F = 0 \] So, the required circle is: \[ x^2 + y^2 -8x -2y = 0 \] Step 3: Use orthogonality condition 
Given other circle: \( x^2 + y^2 + 2x + 2fy = 0 \) If two circles intersect orthogonally, then: \[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \] Here, first circle has: \( g_1 = -4, f_1 = -1, c_1 = 0 \) 
Second circle has: \( g_2 = 1, f_2 = f, c_2 = 0 \) Apply the formula: \[ 2(-4)(1) + 2(-1)(f) = 0 \Rightarrow -8 - 2f = 0 \Rightarrow f = -4 \]  \[ \boxed{f = -4} \]