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Question 18 : If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

- 64
- -32
- 16
- 48

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Given that the sum of squares of two numbers is 97 i.e. a^{2} + b^{2} = 97

From the given options we have to find which one cannot be their product i.e. ab

A. 64 ⟹ 2ab = 128

B. −32 ⟹ 2ab = -64

C. 16 ⟹ 2ab = 32

D. 48 ⟹ 2ab = 96

2ab is found because we know that

a^{2} + b^{2} + 2ab ≥ 0

a^{2} + b^{2} - 2ab ≥ 0

By this we can know that 97 + 128 works but 97 - 128 doesn’t works so we can understand option A cannot be the product and the rest can be.

a^{2} + b^{2} ≥ |2ab|

a^{2} + b^{2} ≥ 2ab

a^{2} + b^{2} ≥ -2ab

⟹ \\frac{a^2 + b^2}{2}) ≥ |ab|

So here 2ab should lie between +97 and -97 or ab should be less than \\frac{97}{2}) or greater than −\\frac{97}{2}), so except option A all the other options works so option A 64 cannot be the product

The question is **"If the sum of squares of two numbers is 97, then which one of the following cannot be their product?"**

Choice A is the correct answer.

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