Question

a,b,c. Multumesc frumos!

Answer 1

a)

$2n-2=n^2-n\Rightarrow n^2-3n+2=0\Rightarrow n^2-n-2n+2=0\Rightarrow n(n-1)-2(n-1)=0\Rightarrow (n-1)(n-2)=0\Rightarrow n-1=0\text{ sau }n-2=0\Rightarrow n=1\text{ sau }n=2\\\text{Deci, punctele }A_1\text{ \c si }A_2\text{ au coordonatele egale.}$

b)

$A_2(2,2),\, A_3(4,6)\\\\\displaystyle A_2A_3:\frac{x-x_{A_3}}{x_{A_2}-x_{A_3}}=\frac{y-y_{A_3}}{y_{A_2}-y_{A_3}}\Rightarrow \frac{x-4}{2-4}=\frac{y-6}{2-6}\Rightarrow\frac{x-4}{-2}=\frac{y-6}{-4}\\\\\Rightarrow-4x+16=-2y+12\Rightarrow A_2A_3:4x-2y-4=0$

c)

$A_1(1,1),A_2(2,2),A_3(4,6)\\\\\displaystyle\mathcal{A}_{A_1A_2A_3}=\frac12|\Delta|=1\\\\\\\Delta=\left|\begin{array}{ccc}x_{A_1} &y_{A_1} &1\\x_{A_2}& y_{A_2}&1\\x_{A_3}&y_{A_3}&1\end{array}\right|=\left|\begin{array}{ccc}1&1&1\\2&2&1\\4&6&1\end{array}\right|=2$