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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?

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Explanatory Answer

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Explanatory Answer

Method of solving this CAT Question from Number theory

Given that the sum of squares of two numbers is 97 i.e. a² + b² = 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
a² + b² + 2ab ≥ 0
a² + b² - 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.
a² + b² ≥ |2ab|
a² + b² ≥ 2ab
a² + b² ≥ -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