Question

if sec²theta = 4xy/(x+y)² holds for some pair (x,y) then how many pairs (x,y) are possible , x,y being real and x is not equal to -y?

Answer 1

sec²A ≥ 1

4xy/(x+y)² ≥ 1
4xy ≥ (x+y)²
4xy ≥ x² + 2xy+y²
0 ≥ x² - 2xy + y²
0 ≥ (x-y)²
Since the square of any real value is non negative, the only case in which this expression can be true is the case of (x-y)² = 0. Therefore sec²A = 4xy/(x+y)² can be true only for the case (a) x = y