Analytical Geometry!

Draw a tangent line of parabola $y=x^2$ at the point $A(1,1)$. Suppose the line intersects the $x$-axis and $y$-axis at $D$ and $B$ respectively. Let point $C$ be on the parabola and point $E$ on $AC$ such that $\dfrac{AE}{EC}=\lambda_{1}$. Let point $F$ be on $BC$ such that $\dfrac{BF}{FC}=\lambda_{2}$ and $\lambda_{1}+\lambda_{2}=1$. Assume that $CD$ intersects $EF$ at point $P$. When point $C$ moves along the parabola, the equation of the trail of $P$ can be expressed in the form $\dfrac{1}{a}(mx+b)^2$. Where $gcd(m, b)=1$, find the value of $a+b+m$.

• This problem is not original.