# Analytical Geometry!

Algebra Level 5

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.
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