Question

Sa se verifice dacă punctele sunt colineare
a). A(-2;1) , B(-1;2) si C (2;-1)
b). M(1;2) , N(-1;4) si P (1;2)
c). E(2;1) , F(-1;3) si Q (2;3)

Answer 1

$a)A(-2,1),B(-1,2),C(2,-1)$

$A(x_{A},y_{A}),B(x_{B},y_{B}),C(x_{C},y_{C})$

$A,B,C \: coliniare \: daca \: :$

$\begin{vmatrix} x_{A}& y_{A} & 1\\ x_{B}& y_{B} & 1\\ x_{C} & y_{C} & 1 \end{vmatrix} = 0$

$\begin{vmatrix} - 2& 1& 1\\ - 1& 2& 1\\ 2& - 1& 1 \end{vmatrix} = 0$

$-2 \times 2 \times 1+(-1) \times (-1) \times 1+2 \times 1 \times 1-1 \times 2 \times 2-1 \times (-1) \times (-1)-1 \times 1 \times (-1) = 0$

$- 4 + 1 + 2 - 4 - 1 + 1 = 0$

$-5 = 0 \: (F)=>A,B,C \:necoliniare$

$b)M(1,2),N(-1,4),P(1,2)$

$M(x_{M},y_{M}),N(x_{N},y_{N}),P(x_{P},y_{P})$

$M,N,P \: coliniare \: daca :$

$\begin{vmatrix} x_{M}& y_{M} & 1\\ x_{N}& y_{N} & 1\\ x_{P} & y_{P} & 1 \end{vmatrix} = 0$

$\begin{vmatrix} 1& 2 & 1\\ - 1& 4& 1\\ 1 & 2 & 1 \end{vmatrix} = 0$

$1 \times 4 \times 1+(-1) \times 2 \times 1+1 \times 2 \times 1-1 \times 4 \times 1-1 \times 2 \times 1-1 \times 2 \times (-1) = 0$

$4 - 2 + 2 - 4 - 2 + 2 = 0$

$0 = 0 \: (A)=>M,N,P\: coliniare$

$c)E(2,1),F(-1,3),Q(2,3)$

$E(x_{E},y_{E}),F(x_{F},y_{F}),Q(x_{Q},y_{Q})$

$E,F,Q \: coliniare \: daca :$

$\begin{vmatrix} x_{E}& y_{E} & 1\\ x_{F}& y_{F} & 1\\ x_{Q} & y_{Q} & 1 \end{vmatrix} = 0$

$\begin{vmatrix} 2& 1 & 1\\ - 1& 3& 1\\ 2 & 3 & 1 \end{vmatrix} = 0$

$2 \times 3 \times 1+(-1) \times 3 \times 1+2 \times 1 \times 1-1 \times 3 \times 2-1 \times 3 \times 2-1 \times 1 \times (-1) = 0$

$6 - 3 + 2 - 6 - 6 + 1 = 0$

$- 6 = 0 \:(F)=>E,F,Q \:necoliniare$