Question

Angles made with the x - axis by the two
lines drawn through the point (1, 2) and

cutting the line x+y=4 at a distance
√2÷3
from the point (1,2) are​

Answer 1

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Let P≡(1,2)

Using the parametric form , the coordinates of the point at a distance of

$\sqrt{ \frac{2}{3} }$

is given by

$1 + \sqrt{ \frac{2}{3} } \cosθ</p><p></p><p> ,</p><p></p><p>2 + \sqrt{ \frac{2}{3} } \sinθ$

This point lies on the line x+y=4

Hence,

$3 + \sqrt{ \frac{2}{3} } (\cosθ</p><p></p><p> + \sinθ ) = 4$

On squaring and simplifying, we get

${ \sin }^{2} θ</p><p></p><p> + { \cos }^{2} θ</p><p></p><p> + 2 \: \sinθ</p><p></p><p> \cosθ = \frac{3}{2} </p><p></p><p>$

$1 + \sin2θ</p><p></p><p> = \frac{2}{3}$

$\sin2θ</p><p></p><p> = \frac{1}{2}$

the possible value of θ are

$\frac{\pi}{12}$

and

$\frac{5\pi}{12}$