Question

In sistemul cartezian xOy se considera punctele An(5n, 8n+1), n apartinand lui N.
a. Scrieti ecuatia dreptei A1A2
b. Verificati daca punctele A1,A2,An sunt coliniare, n apartinand lui N, n>=3.

Answer 1

a)

$A_1(5,9),\,A_2(10, 17)\\A_1A_2:\displaystyle\frac{x-x_2}{x_1-x_2}=\frac{y-y_2}{y_1-y_2}\Rightarrow\frac{x-10}{5-10}=\frac{y-17}{9-17}\Rightarrow A_1A_2:8x-5y-5=0$

b)

$A_1,A_2,A_n\text{ coliniare}\Leftrightarrow\Delta=0\\\displaystyle\Delta=\left|\begin{array}{ccc}x_1 &y_1 &1\\x_2 &y_2 &1\\x_n &y_n &1\end{array}\right|=\left|\begin{array}{ccc}5 &9 &1\\10 &17 &1\\5n &8n+1 &1\end{array}\right|=85+80n+10+45n-85n-40n-5-90=0\Rightarrow A_1,\,A_2,\,A_n\text{ coliniare},\,\forall n\in\mathbb{N},\,n\geq3$