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Let a, b, c be positive integers such th...

Let a, b, c be positive integers such that `(asqrt2+b)/(bsqrt2+c)` is a rational number, then which of the following is always an integer?

A

`(2a^(2)+b^(2))/(2b^(2)+c^(2))`

B

`(a^(2)+2b^(2))/(b^(2)+2c^(2))`

C

`(a^(2)+b^(2)-c^(2))/(a+b+c)`

D

`(a^(2)+b^(2)+c^(2))/(a+b-c)`

Text Solution

Verified by Experts

The correct Answer is:
D
`((asqrt2+b)/(bsqrt2+c))((bsqrt2-c)/(bsqrt2-c))=(2ab-sqrt2ac+sqrt2b^(2)-bc)/(2b^(2)-c^(3))=(b(2a-c)+sqrt2(b^(2)-ac))/(2b^(2)-c^(2))`
Is rational when `b^(2)` ac i.e., a, b, c are in GP
Here given option (A), (B), (C) does not satisfy the criteria
But option (D) always satisfy
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