Question

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.

Answer 1

Hint: In this question three coordinates are being told to be collinear and a relation between a and b is given to us, we need to find the value of a and b. If some points are collinear this means that these points must lie on the same line thus slopes must be equal, use this concept along with the given relation between a and b to get the answer.

Complete step-by-step answer:
Given points are

${\text{A}}\left( { - 2,1} \right),{\text{ B}}\left( {a,b} \right),{\text{ and C}}\left( {4 - 1} \right)$

Now we know that three points are collinear if their slopes are equal

Therefore slope of AB $ = $ Slope of BC

Collinearity of points: - Collinear points always lie on the same line.

Now we know

Slope between two points ${\text{ = }}\left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)$

Consider $A\left( { - 2,1} \right) \equiv \left( {{x_1},{y_1}} \right),{\text{ }}B\left( {a,b} \right)

\equiv \left( {{x_2},{y_2}} \right),{\text{ }}C\left( {4, - 1} \right) \equiv \left( {{x_3},{y_3}}

\right)$

Therefore slope of AB${\text{ = }}\left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right){\text{ =

}}\dfrac{{b - 1}}{{a + 2}}$

Therefore slope of BC${\text{ = }}\left( {\dfrac{{{y_3} - {y_2}}}{{{x_3} - {x_2}}}} \right){\text{ =

}}\dfrac{{ - 1 - b}}{{4 - a}}$

Points are collinear

Therefore slope of AB $ = $ Slope of BC

$

   \Rightarrow \dfrac{{b - 1}}{{a + 2}} = \dfrac{{ - 1 - b}}{{4 - a}} \\

   \Rightarrow \left( {b - 1} \right)\left( {4 - a} \right) = \left( {a + 2} \right)\left( { - 1 - b} \right)

\\

   \Rightarrow 4b - ab - 4 + a = - a - ab - 2 - b \\

$

$ \Rightarrow 5b + 2a = 2$………………….. (1)

Now it is also given that

$\left( {a - b} \right) = 1$…………………….. (2)

Now from equation (2) the value of a is

$ \Rightarrow a = 1 + b$

Now put this value in equation (1) we have,

$ \Rightarrow 5b + 2\left( {1 + b} \right) = 2$

Now simplify this equation we have,

$

   \Rightarrow 5b + 2 + 2b = 2 \\

   \Rightarrow 7b = 0 \\

   \Rightarrow b = 0 \\

 $

Now put this value in equation (2) we have,

$

   \Rightarrow \left( {a - 0} \right) = 1 \\

   \Rightarrow a = 1 \\

$

So, the value of a and b is 1 and 0 respectively.

Note: Whenever we face such types of problems the key concept is simply to have the gist of understanding of the physical interpretation of points being collinear. Knowledge of the basic slope formula also helps in getting on the right track to get the answer.