Question

If \[ x^2+y^2-2x+2fy+c=0 \] intersects the two circles \[ x^2+y^2+2x-4y+1=0 \] and \[ x^2+y^2-4x-2y-11=0 \] orthogonally, then \[ f+c= \]

• A. \(20\)
• B. \(17\)
• C. \(12\)
• D. \(24\) \bigskip

Hint:

For orthogonal circles, \[ 2g_1g_2+2f_1f_2=c_1+c_2. \] This relation is often much faster than converting the circles into centre-radius form.

Answer 1

The Correct Option is D

Concept: Two circles \(x^2+y^2+2g_1x+2f_1y+c_1=0\) and \(x^2+y^2+2g_2x+2f_2y+c_2=0\) cut orthogonally if \(2g_1g_2+2f_1f_2=c_1+c_2\). Apply this condition with both given circles to find \(f\) and \(c\), then sum them.

Step 1: Required circle: \(x^2+y^2-2x+2fy+c=0 \Rightarrow g=-1, f_?=f\). First given circle: \(x^2+y^2+2x-4y+1=0 \Rightarrow g_1=1, f_1=-2, c_1=1\). Orthogonality: \(2(-1)(1)+2(f)(-2)=c+1 \Rightarrow -2-4f=c+1 \Rightarrow c=-3-4f\). (1)

Step 2: Second given circle: \(x^2+y^2-4x-2y-11=0 \Rightarrow g_2=-2, f_2=-1, c_2=-11\). Orthogonality: \(2(-1)(-2)+2(f)(-1)=c-11 \Rightarrow 4-2f=c-11 \Rightarrow c=15-2f\). (2)

Step 3: Equate (1) and (2): \(-3-4f=15-2f \Rightarrow -2f=18 \Rightarrow f=-9\). Then \(c=15-2(-9)=33\).

Step 4: Find \(f+c = -9+33 = 24\).

Step 5: Write the final answer. \(\boxed{24}\)