First slide

Equation of line in Straight lines

Question

Angles made with the $x -$ axis by two lines drawn through the point $(1, 2)$ and cutting the line $x + y = 4$ at a distance $\frac{\sqrt{6}}{3}$ from the point $(1, 2)$ are

Moderate

A

$π / 6$ and $π / 3$
B

$π / 8$ and $3 π / 8$
C

$π / 12$ and $5 π / 12$
D

none of these

Solution

Suppose a line passing through $(1, 2)$ makes an angle $θ$ with $x -$ axis. Then, its equation is
$\frac{x - 1}{\cos θ} = \frac{y - 2}{\sin θ}$
The coordinates of a point on this line at a distance $\frac{\sqrt{6}}{3} = \sqrt{\frac{2}{3}}$ from $(1, 2)$ are given by
$\frac{x - 1}{\cos θ} = \frac{y - 2}{\sin θ} = \sqrt{\frac{2}{3}}$ or, $(1 + \sqrt{\frac{2}{3}} \cos θ, 2 + \sqrt{\frac{2}{3}} \sin θ)$
This point lies on $x + y = 4 .$
$∴ 1 + \sqrt{\frac{2}{3}} \cos θ + 2 + \sqrt{\frac{2}{3}} \sin θ = 4$
$\Rightarrow \cos θ + \sin θ = \sqrt{\frac{3}{2}}$
$\Rightarrow \sin (θ + \frac{π}{4}) = \frac{\sqrt{3}}{2} \Rightarrow θ + \frac{π}{4} = \frac{π}{3}, \frac{2 π}{3} \Rightarrow θ = \frac{π}{12}$ and $\frac{5 π}{12}$

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