Question

The equation of a circle is $Re \left(z^{2}\right)+2(Im(z))^{2}+2Re(z)=0$, where $z=x+ i y$. A line which passes through the center of the given circle and the vertex of the parabola, $x^{2}-6 x-y+13=0$, has $y$-intercept equal to______.

Answer 1

Equation of circle is $\left(x^{2}-y^{2}\right)+2 y^{2}+2 x=0$
$x^{2}+y^{2}+2 x=0$
Centre: $(-1,0)$
Parabola $: x^{2}-6 x-y+13=0$
$(x-3)^{2}=y-4$
Vertex: $(3,4)$
Equation of line $\equiv y-0=\frac{4-0}{3+1}(x+1)$
$y=x+1$
$y$-intercept $=1$