⇒|z|2+2⋅Re(az)−4=0.....(i) [z+z=2Re(z)] Now, given z=x+iy⇒|z|2=x2+y2 and a=1+i, then az=(1+i)(x+iy)
∴az=(x−y)+i(x+y)⇒Re(az)=x−y
∴ From Eq. (i), put Re(az)=x−y and |z|2=x2+y2 ⇒x2+y2+2(x−y)−4=0 ⇒S:x2+y2+2x−2y−4=0...(i) is a circle and given line L:z+z−i(z−z)+2=0 ⇒L:2Re(z)−i2Im(z)+2=0 ⇒L:2x−i(2yi)+2=0 ⇒L:x+y+1=0...(iii) Now, equation of circle passing through point of intersection of circle S and line L is S+λL=0
(x2+y2+2x−2y−4)+λ(x+y+1)=0...(iv)
Pass through (0,0) ∴ Put in Eq. (iv), we get −4+λ=0⇒λ=4 Now, required circle is