Lines $y=3x+4$ and $y=mx+1$ intersect parabola $x^2+x-y+1=0$ at four distinct concyclic points. If possible values of $m$ are $m_i$, find $\sum m_i^2$:
Step-by-Step Solution
Key Concept: $y=3x+4$ cuts parabola at $A(-1,1)$ and $B(3,13)$. $y=mx+1$ cuts at $C(0,1)$ and $D(m-1,m^2-m+1)$. Circle through $A,B,C$: $x^2+y^2+x-15y+14=0$. Require $D$ on this circle.
$\sum m_i^2=\mathbf{9}$ (only $m=-3$ valid, giving $D(-4,13)$ distinct from $A,B,C$).
Correct Answer: 9