First slide

Equation of line in Straight lines

Question

The set of all possible values of $θ$ in $(0, π)$ for which the points $P (1, 2)$ and $Q (\sin θ, \cos θ)$ lie on the same side of the lines $x + y = 1,$ is

Moderate

A

$(0, π / 2)$
B

$(0, π / 4)$
C

$(π / 4, 3 π / 4)$
D

$(0, 3 π / 4)$

Solution

If points $P (1, 2)$ and Q $(\sin θ, \cos θ)$ lie on the same side of the line $x + y - 1 = 0$ , then
$(1 + 2 - 1) (\sin θ + \cos θ - 1) > 0$
$\begin{array}{l} \Rightarrow \sin θ + \cos θ > 1 \\ \Rightarrow \sqrt{2} \sin (θ + \frac{π}{4}) > 1 \\ \Rightarrow \sin (θ + \frac{π}{4}) > \frac{1}{\sqrt{2}} \Rightarrow \frac{π}{4} < θ + \frac{π}{4} < \frac{3 π}{4} \Rightarrow 0 < θ < \frac{π}{2} \end{array}$

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