First slide

Equation of line in Straight lines

Question

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

Moderate

A

$(0, \frac{π}{2})$
B

$(0, \frac{π}{4})$
C

$(\frac{π}{4}, \frac{π}{2})$
D

$(\frac{π}{4}, 0)$

Solution

L₁₁ and L₂₂ have same sign since $L_{11} = 3 + 5 - 1 > 0, ∴ L_{22} = \sin θ + \cos θ - 1 > 0$
$\Rightarrow$ $\sin θ + \cos θ > 1 \Rightarrow \sin (\frac{π}{4} + θ) > \frac{1}{\sqrt{2}}$
$\Rightarrow$ $\sin (\frac{π}{4} + θ) > \sin \frac{π}{4} \Rightarrow \frac{π}{4} < θ + \frac{π}{4} < \frac{3 π}{4}$
$\Rightarrow$ $0 < θ < \frac{π}{2} .$ Hence $θ \in (0, \frac{π}{2})$ .

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