is the real root and , are the other roots of the equation x^3-a^3=0(a>0) , then the number of common points of the curves given by |z- |= 3 a 2 and |z- |= 3 2 is

Step-by-step solution

x^3-a^3=0 (x-a) (x^2+a^2+a x )=0 x=a, -a a^2-4 a^2 2 aligned & x=a, -a 2 i a 3 2 & x=a is a real root. So, a= & Now, = -a 2 +i a 3 2 & r= -a 2 -i a 3 2 & |z- |= 3 a 2 |x+i y+ a 2 -i a 3 2 |^2= ( 3 2 a )^2 & (x+ a 2 )^2+ (y- 3 2 a )^2= 3 a^2 4 aligned x^2+y^2+a x- 3 a y+ a^2 4 =0 ...(i) aligned & Also, |z-r|= 3 2 |x+i y+ a 2 +i a 3 1 |^2= ( 3 2 )^2 & (x+ a 2 )^2+ (y+ a 3 2 )^2= 3 4 ^2 aligned x^2+y^2+a x+ 3 a y+ a^2 4 =0 ...(ii) Equations (i) & (ii): aligned & 2 3 a y=0 y=0 & From eq ^ n ( ii ) x^2+0+a x+0+ a^2 4 =0