The equation of a circle is Re z 2 + 2 Im z 2 + 2 Re z = 0 , where z = x + i y . A line which passes through…

We have, $Re\left(z^2\right)+2{\left(Im\left(z\right)\right)}^2+2Re\left(z\right)=0$

where $z=x+iy$

Equation of circle is $\left(x^2-y^2\right)+2y^2+2x=0$

$x^2+y^2+2x=0$

Centre: $\left(-1,0\right)$

Parabola : ${\left(x-3\right)}^2=y-4$

Vertex : $\left(3,4\right)$

Equation of line $\equiv y-0=\frac{4-0}{3+1}\left(x+1\right)$

$\Rightarrow y=x+1$

Hence, $y-$  intercept $=1.$