6. On the plane $x O y$, the lines $y=3 x-3$ and $x=-1$ intersect at point $\mathrm{B}$, and the line passing through point $M(1 ; 2)$ intersects the given lines at points A and C respectively. For what positive value of the abscissa of point A will the area of triangle $\mathrm{ABC}$ be the smallest?

(12 points)
# Solution.

$A C: \quad y=k x+d, \quad M \in A C \Rightarrow d=2-k$

$A(a ; 3 a-3) \in A C \Rightarrow 3 a-3=k a+2-k \Rightarrow a=\frac{5-k}{3-k}$,

$C(-1 ; c) \in A C \Rightarrow c=-2 k+2$,

$S_{A B C}=\frac{1}{2}(c+6) \cdot(a+1)=\frac{2(k-4)^{2}}{3-k}$

$S^{\prime}=\frac{2(k-4)(2-k)}{(3-k)^{2}}=0, k_{\min }=2, \quad a_{\text {min }}=3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e353629715b7c8c09e0g-02.jpg?height=734&width=400&top_left_y=1578&top_left_x=1496)

Answer: 3.