First slide

Equation of line in Straight lines

Question

The range of $θ$ in $(0, π)$ such that the point (3,5) and $(\sin θ, \cos θ)$ lie on the same side of the line x + y – 1 = 0 is

Moderate

A

$(0, \frac{π}{4})$
B

$(0, \frac{π}{2})$
C

$(\frac{π}{4}, \frac{3 π}{4})$
D

$(\frac{π}{2}, \frac{3 π}{4})$

Solution

$3 + 5 - 1 > 0$
$\sin θ + \cos θ > 1$ (divide with $\sqrt{2}$ )
$\sin (θ + \frac{π}{4}) > \frac{1}{\sqrt{2}} = \sin \frac{π}{4}$ $(T h e p o i n t s A, B a r e o n s a m e s i d e o f a l i n e L \equiv 0 t h e n L_{A}, L_{B} h a v e s a m e s i g n)$
$\Rightarrow$ $θ + \frac{π}{4} \in (\frac{π}{4}, \frac{3 π}{4})$
$\Rightarrow$ $θ \in (0, \frac{π}{2})$

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