Question

O valor de x para que os pontos A(-2, 5), B(4, -3) e C(x, 1) sejam colineares é:

Answer 1

Podemos utilizar o método do determinante. Se o determinante formado pelos pontos (ver abaixo) for igual a zero, os pontos são colineares.

$\left|\begin{array}{ccc}x_A&y_A&1\\x_B&y_B&1\\x_C&y_C&1\end{array}\right|~=~0\\\\\\\\\left|\begin{array}{ccc}-2&5&1\\4&-3&1\\x&1&1\end{array}\right|~=~0\\\\\\\\(~(-2).(-3).1+4.1.1+x.5.1~)~-~(~1.(-3).x+1.1.(-2)+1.5.4~)~=~0\\\\\\(~6+4+5x~)~-~(~-3x-2+20~)~=~0\\\\\\(5x+10)~-~(-3x+18)~=~0\\\\\\5x+10+3x-18~=~0\\\\\\8x-8~=~0\\\\\\8x~=~8\\\\\\x~=~\frac{8}{8}\\\\\\\boxed{x~=~1}$