Question

The objective function of a LPP is given by \(z = ax + by\), where \(a\) and \(b\) are constants. If minimum of \(z\) occurs at two points \((50, 30)\) and \((20, 40)\), then

• A. \(3b = a\)
• B. \(3b + a = 0\)
• C. \(3a + b = 0\)
• D. \(3a - b = 0\)

Hint:

Equal minimum at two points means \(50a+30b=20a+40b\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In a linear programming problem, the objective function \(z=ax+by\) can reach its minimum at two corner points only when both points give the same value of \(z\). Then every point on the segment joining them gives that same minimum.

Step 2: Key Formula or Approach:
Put the two points into \(z\) and set the two values equal.

Step 3: Substitute the points.
At \((50,30)\): \(z=50a+30b\).
At \((20,40)\): \(z=20a+40b\).

Step 4: Equate and simplify.
\[ 50a+30b=20a+40b \]
\[ 30a = 10b \]
\[ 3a = b \quad\Rightarrow\quad 3a-b=0 \]

Step 5: Check the options.
Option 1 says \(a=3b\), which is the reverse ratio. Option 2 says \(a=-3b\), which has the wrong sign. Option 3 says \(b=-3a\), also the wrong sign. Only \(3a-b=0\) matches.

Final Answer:
The condition is \(3a-b=0\), which is option 4. \[ \boxed{3a - b = 0} \]