The two circles x^2 + y^2 = ax and x^2 + y^2 = c^2

The two circles x2 + y2 = ax and x2 + y2 = c2 (c > 0) touch each other if

  1. a = 2c
  2. |a| = 2c
  3. |a| = c
  4. 2|a| = c

Answer

x2 + y2 = ax

Centre c1(-a/2,0) and radius r1 = |a/2|

x2 + y2 = c2

Centre c2(0,0) and radius r2 = c

Both touch each other iff

|c1 + c2| = r1 ± r2

a2/4 = (±a/2±c)2 

a2/4 = a2/4 ± |a|c + c2

The correct option is C.