If the sum of squares of two numbers is 97, then which one of the following cannot be their product?
Started 3 months ago by Shashank in
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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