Question

Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.

Answer 1

The formula for the area ‘A’ encompassed by three points, and is given by the formula,

$∆=\frac{1}{2}\left|\left(x_1{y}_2+x_2{y}_3+x_3{y}_1\right)-\left(x_2{y}_1+x_3{y}_2+x_1{y}_3\right)\right|$

If three points are collinear the area encompassed by them is equal to 0.

The three given points are A(8,1), B(3,−4) and C(2,k). It is also said that they are collinear and hence the area enclosed by them should be 0.
$$\begin{gathered} ∆=\frac{1}{2}\left|\left(8\times -4+3\times k+2\times 1\right)-\left(3\times 1+2\times -4+8\times k\right)\right| \\ 0=\frac{1}{2}\left|\left(-32+3k+2\right)-\left(3-8+8k\right)\right| \\ 0=\frac{1}{2}\left|-25-5k\right| \\ k=-5 \end{gathered}$$

Hence the value of ‘k’ for which the given points are collinear is.