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Find 2ab from a⁴+a²b²+b⁴=900 and a²+ab+b²=45 | IOQM

JEE Maths question with a full step-by-step solution.

Question

Let

a

and

b

be real numbers such that

a^4 + a^2 b^2 + b^4 = 900

and

a^2 + ab + b^2 = 45

. Find the value of

2ab

Solution

Answer: 25

Step 1: Record the given relations.

a^2 + ab + b^2 = 45 \quad \text{...(i)}, \qquad a^4 + a^2 b^2 + b^4 = 900.

Step 2: Factor the quartic, using

a^4 + a^2 b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)

(a^2 + ab + b^2)(a^2 - ab + b^2) = 900.

Step 3: Substitute (i):

45(a^2 - ab + b^2) = 900 \implies a^2 - ab + b^2 = 20 \quad \text{...(ii)}.

Step 4: Subtract (ii) from (i):

(a^2 + ab + b^2) - (a^2 - ab + b^2) = 45 - 20 \implies 2ab = 25.

Answer:

2ab = 25

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