Let the locus of a point z in the Argand plane satisfying the condition Re ( z ^2 )=4 be C ₁ and the locus of z satisfying the condition Im ( z ^2 )=4 be C ₂ . Then the number of common points of the two curves C ₁ and C ₂ are

Step-by-step solution

Let z=x+i y z^2=x^2-y^2+i 2 x y aligned & Re (z^2 )=x^2-y^2=4 ..(i) & Im (z^2 )=2 x y=4 x y=2 ...(ii) aligned aligned & from eq ^ n (i) and (ii) x ^2- 4 x ^2 =4 & x ^4-9 x ^2-9=0 aligned x^2= 4 16+16 2 =2 2 2 aligned & x^2=2(1 2 ) & x^2=2(1- 2 ) < 0 (Not possible). aligned x^2=2(1+ 2 ) x= [2(1+ 2 )]^ 1 2 So, y= 2 [2(1+ 2 )]^ 1 2 We get 2 common points.