CUET Mathematics 2024 - The corner points of the feasible region determined by x+y ≤ 8, 2 x+y ≥ 8, x ≥ 0, y ≥ 0 are A(0,8), B(4,0) and C(8,0). If the objective function Z=a x+ by has its maximum value on the line segment A B, then the relation between a and b is : | PYQs + Solutions

CUET Mathematics 2024

Algebra

Linear Programming

Medium

The corner points of the feasible region determined by $x+y \leq 8, 2 x+y \geq 8, x \geq 0, y \geq 0$ are $A(0,8), B(4,0)$ and $C(8,0)$ . If the objective function $Z=a x+$ by has its maximum value on the line segment $A B$ , then the relation between $a$ and $b$ is :

8 a+4=b

a=2 b

\mathrm{b}=2 \mathrm{a}

8 b+4=a

✅ Correct Option: 2

Let's calculate the value of

Z

at each corner point:

At point

A(0,8)

Z_A = a(0) + b(8) = 8b

At point

B(4,0)

Z_B = a(4) + b(0) = 4a

At point

C(8,0)

Z_C = a(8) + b(0) = 8a

For a maximum to occur on segment

AB

(not just at endpoints), the function must increase from

C

B

and from

B

A

For

Z

to increase from

C

B

Z_B > Z_C

which means

4a > 8a

, so

a < 0

For

Z

to increase from

B

A

Z_A > Z_B

which means

8b > 4a

, so

2b > a

For the maximum to occur exactly on segment

AB

, the level curves of

Z

must be parallel to

AB

The slope of

AB

is:

\dfrac{0-8}{4-0} = \dfrac{-8}{4} = -2

The slope of the level curves

Z=ax+by

-\dfrac{a}{b}

For these to be parallel:

-\dfrac{a}{b} = -2

Which simplifies to:

a = 2b

The relation between

a

and

b

a = 2b

We can verify: If

a = 2b

, then

Z_A = 8b

and

Z_B = 4a = 4(2b) = 8b

, confirming equal values at both points.