If the radical axis of the circles (x^2+y^2+2 g x+2 f y+c=0 ) and (2 x^2+2 y^2+3 x+8 y+2 c=0 ) touches the circle (x^2+y^2+2 x+2 y+1=0 ), then ((4 g-3)(f-2)= )

Step-by-step solution

The radical axis of the circles ( aligned & x^2+y^2+2 g x+2 f y+c=0 & and 2 x^2+2 y^2+3 x+8 y+2 x=0 & is (4 g-3) x+(4 f-8) y=0 (i) aligned ) Since, the radical axis (i) touches the circle ( aligned & x^2+y^2+2 x+2 y+1=0, so & |-(4 g-3)-(4 f-8)| (4 g-3)^2+(4 f-8)^2 = 1+1-1 & (4 g-3)^2+(4 f-8)^2+2(4 g-3)(4 f-8) & =(4 g-3)^2+(4 f-8)^2 & 8(4 g-3)(f-2)=0 & (4 g-3)(f-2=0 aligned ) Hence, option (1) is correct.