[laTEX](d) : y = m(x-1) = mx-m \\ \\ x+ 2 = (mx-m)(x-1) \\ \\ mx^2 -(2m+1)x+m -2 =0 \\ \\ m \not = 0 \\ \\ \Delta = 4m^2+4m+1 +8m = 4m^2+12m+1 > 0 \\ \\ x_1 \not = x_2 \\ \\ x_1+x_2 = \frac{2m+1}{m} \\ \\ x_1.x_2 = \frac{m-2}{m} \\ \\ M (x_1,mx_1-m) \\ \\ N (x_2,mx_2-m) \\ \\ \vec{AM} = ( x_1-1, m.x_1-m) \\ \\ \vec{AN} = (x_2-1,mx_2 -m) \\ \\ AM^2 = 4AN^2 \\ \\ (m^2+1).(x_1-1)^2 = 4(m^2+1)(x_2-1)^2 [/laTEX]
[laTEX] TH_1: x_1-1 = 2(x_2-1) \\ \\ x_1 -2x_2 = - 1 \\ \\ \begin{cases} x_1+x_2 = \frac{2m+1}{m} \\ x_1 -2x_2 = - 1 \\ x_1.x_2 = \frac{m-2}{m} \end{cases} \\ \\ \Rightarrow \frac{(3m+2)(3m+1)}{9m^2} = \frac{m-2}{m} \\ \\ \frac{9m^2+9m +2 }{9m} = m -2 \\ \\ 9m^2 +9m +2 = 9m^2 -18m \\ \\ m = -\frac{2}{27} \Rightarrow x_1 = -8 < 1 \\ \\ x_2 = -3,5 < 1 (L)[/laTEX]
[laTEX] TH_2: x_1-1 = -2(x_2-1) \\ \\ x_1 +2x_2 = 3 \\ \\ \begin{cases} x_1+x_2 = \frac{2m+1}{m} \\ x_1 +2x_2 = 3 \\ x_1.x_2 = \frac{m-2}{m} \end{cases} \\ \\ \Rightarrow \frac{(m-1)(m+2)}{m^2} = \frac{m-2}{m} \\ \\ \frac{m^2+m-2 }{m} = m -2 \\ \\ m^2+m-2 = m^2-2m \\ \\ m = \frac{2}{3} \Rightarrow x_1 = 4 > 1 \\ \\ x_2 = -\frac{1}{2} <1 (T/M) [/laTEX]
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