Question 147
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 $\frac{AE}{EC} = \lambda_1$. Let point $F$ be on $BC$ such that $\frac{BF}{FC} = \lambda_2$ and $\lambda_1 + \lambda_2 = 1$. Assume that $CD$ intersects $EF$ at point $P$. When point $C$ moves along the parabola, find the equation of the trail of $P$.