Find the smallest positive values of `xa n dy` satisfying `x-y=pi/4a n dcotx+coty=2`

Options

A. `x = (pi)/(6), y = (5pi)/(12)`
B. `x = (5pi)/(12), y = (pi)/(6)`
C. `x = (pi)/(3), y=(7pi)/(12)`
D. none of these

Answer

B

Explanation

We have,
`x-y = (pi)/(4)`
`"and, cot"x + "cot"y =2`
Now,
`"cot"x + "cot"y = 2`
`rArr "cot" (y + (pi)/(4)) + "cot" y = 2`
`rArr ("cot" y-1)/("cot"y +1) + "cot" y = 2`
`rArr "cot"^(2)y + 2 "cot" y-1 = 2 "cot" y + 2`
`rArr "cot"^(2)y = 3`
`rArr y = (pi)/(6) " " [because y " is the smallest positive angle"]`
`rArr x = y + (pi)/(4) = (pi)/(6) + (pi)/(4) = (5pi)/(12)`
Hence, `x = (5pi)/(12)" and " y = (pi)/(6)`