Question

If the points X(a, b + c), Y(b, a + c) and Z(k, a + b) are collinear, then find the value of k. [2 marks]

Answer 1

If the three points X, Y and Z are collinear, then, the area of triangle XYZ is zero. [1 mark]

$\Rightarrow \frac{1}{2}|a(a+c-a-b)+b(a+b-b-c)+k(b+c-a-c)|=0$

$\Rightarrow \frac{1}{2}|a(c-b)+b(a-c)+k(b-a)|=0$

$\Rightarrow \frac{1}{2}|ac-ab+ba-bc+kb-ka|=0$

$\Rightarrow ac-bc+kb-ka=0$

$\Rightarrow k(b-a)=c(b-a)$

$\Rightarrow k=c$ [1 mark]