Analytical Geometry!

Algebra Level 5

Draw a tangent line of parabola y=x2y=x^2 at the point A(1,1)A(1,1). Suppose the line intersects the xx-axis and yy-axis at DD and BB respectively. Let point CC be on the parabola and point EE on ACAC such that AEEC=λ1\dfrac{AE}{EC}=\lambda_{1}. Let point FF be on BCBC such that BFFC=λ2\dfrac{BF}{FC}=\lambda_{2} and λ1+λ2=1\lambda_{1}+\lambda_{2}=1. Assume that CDCD intersects EFEF at point PP. When point CC moves along the parabola, the equation of the trail of PP can be expressed in the form 1a(mx+b)2\dfrac{1}{a}(mx+b)^2. Where gcd(m,b)=1gcd(m, b)=1, find the value of a+b+ma+b+m.

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