If the points 𝐴(− 2, 1), 𝐵(𝑎, 𝑏) and 𝐶(4, 1) are collinear and 𝑎 − 𝑏 = 1, find 𝑎 and 𝑏.

We need to find the value of 𝑎 and 𝑏 if the points 𝐴(− 2, 1), 𝐵(𝑎, 𝑏) and 𝐶(4, 1)
are collinear and 𝑎 − 𝑏 = 1.
If three points are collinear, then it can be said that the area of the triangle formed by the points is zero. Mathematically, if points 𝑃, 𝑄 and 𝑅 are collinear, then 𝑎𝑟(△𝑃𝑄𝑅) = 0, i.e.,
if coordinates of points 𝑃, 𝑄 and 𝑅 are (𝑥₁ , 𝑦₁ ), (𝑥₂, 𝑦₂ ) and (𝑥₃ , 𝑦₃ ) respectively, then
$\frac{1}{2}\left[x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right] = 0$
 

If we assume 𝑃(𝑥₁, 𝑦₁)= 𝐴(–2, 1), Q(𝑥₂, 𝑦₂)= 𝐵(𝑎, 𝑏) and 𝑅(𝑥₃, 𝑦₃)= 𝐶(4, 1), then on substituting the values in the above relation, we get

$$\begin{gathered} \frac{1}{2}\left[-2(b-1) + a(1-1) + 4(1-b)\right] = 0 \\ \frac{1}{2}\left[-2b + 2 + 4 - 4b\right] = 0 \end{gathered}$$

On multiplying both sides by 2, we get

⇒ [− 2𝑏 + 2 + 4 − 4𝑏] = 0

⇒ − 6𝑏 + 6 = 0

⇒ 𝑏 = 1

Now, it is given that 𝑎 − 𝑏 = 1, therefore, we get

𝑎 − 𝑏 = 1

⇒ 𝑎 − 1 = 1

⇒ 𝑎 = 2

Hence, the value of 𝑎 and 𝑏 are 2 and 1 respectively.