Question

Two vertices of the feasible region P of an LPP are x₁ = (0, 1, 1, 2) and x₂ = (3, 0, 0, 1). Then, a point which can be a vertex of P is
1. (1, 2/3, 2/3, 5/3)
2. (6, -1, -1, 0)
3. (1/2, 5/6, 5/6, 11/6)
4. (1,2/3, 2/3, 5/3), (6, -1, -1, 0), (1/2, 5/6, 5/6, 11/6)

Answer 1

Correct Answer - Option 4 : (1,2/3, 2/3, 5/3), (6, -1, -1, 0), (1/2, 5/6, 5/6, 11/6)

Concept:

Equation of line joining two points (x1, y1) and (x2, y2)

\(y\; - {y_1} = \frac{{\left( {{y_2} - {y_1}} \right)}}{{({x_2} - {x_1})}}\left( {x - {x_1}} \right)\)

Calculation:

Given:

x₁ = (0, 1, 1, 2) and x₂ = (3, 0, 0, 1)

For simplicity, let consider the points x₁ = (0, 1) and x₂ = (3, 0)

Equation of the line joining the above two points is,

\(\left( {y - 1} \right) = \frac{{0 - 1}}{{3 - 0}}\left( {x - 0} \right)\)

\( \Rightarrow 3\left( {y - 1} \right) = - x\)

\( \Rightarrow x + 3y = 3\)

Now check the options:

(1, 2/3), (6, -1), (1/2, 5/6) all these points satisfies the above equation. Hence all these points lies on the same line.

Now we can say that all these points are colli near points. Hence all these points can be vertex of P.