Question

For what value of a the point (a, 1), (1, −1), and (11, 4) 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(a, 1), B(1, −1) and C(11, 4). It is also said that they are collinear and hence the area enclosed by them should be 0.
$$\begin{gathered} ∆=\frac{1}{2}\left|\left(x_1{y}_2+x_2{y}_3+x_3{y}_1\right)-\left(x_2{y}_1+x_3{y}_2+x_1{y}_3\right)\right| \\ 0=\frac{1}{2}\left|\left(a\times -1+1\times 4+11\times 1\right)-\left(1\times 1+11\times -1+a\times 4\right)\right| \\ 0=\frac{1}{2}\left|\left(-a+4+11\right)-\left(1-11+4a\right)\right| \\ 0=\frac{1}{2}\left|\left(-a+15\right)-\left(-10+4a\right)\right| \\ 0=\frac{1}{2}\left|-a+15+10-4a\right| \\ 0=\frac{1}{2}\left|-5a+25\right| \\ 0=-5a+25 \\ 5a=25 \\ a=5 \end{gathered}$$

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