The range of values of `theta` in the interval `(0, pi)` such that the points (3,5) and `(sin theta , cos theta)` lie on the same side of the line `x+y-1=0`, is

Options

A. `0 lt theta lt (pi)/(4)`
B. `0 lt theta lt (pi)/(2)`
C. `0 lt theta lt pi`
D. `(pi)/(4) lt theta lt (3pi)/(4)`

Answer

B

Explanation

We have line x+y-1=0.
Let L(x,y) = x+y-1.
`therefore L(3,5) = 3+5-1=7 gt 0`
For points (3,5) and `"sin" theta, "cos" theta)` to lie on the same side line x+y-1=0, we must have
`L("cos" theta, "sin" theta) gt 0` `"or " "sin" theta + "cos" theta-1 gt 0`
`"sin" ((pi)/(4)+ theta) gt (1)/(sqrt(2))`
`(pi)/(4) lt (pi)/(4) + theta lt (3pi)/(4)`
`0 lt theta lt (pi)/(2)`