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7 months ago
If the sum of squares of two numbers is 97, then which one of the following cannot be their product?
Explanatory Answer
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
⟹( a 2 + b 2 ) / 2 ≥ |ab|
So here 2ab should lie between +97 and -97 or ab should be less than 97/2 or greater than −97/2,
so except option A all the other options works so option A 64 cannot be the product
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