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Q.

A man, at the crossing of two roads $x - 2 y - 4 = 0, 2 x - y - 4 = 0$ starts walking along the bisector of the acute angle between the roads and after covering a distance $\sqrt{2}$ units, reaches the bank of a straight river at right angles to his path L. Which of the following is/ are true

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a

the equation of bank, can be $x + y + 2 = 0$

b

Origin lies in the region of the obtuse angle bisector

c

Equation of acute bisector is $3 x - 3 y - 8 = 0$ , and $x + y = 0$ obtuse bisector is

d

the equation of bank, can be $3 x - 4 y + 5 = 0$

answer is A, C, D.

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Detailed Solution

Point of intersection of the roads is $(\frac{4}{3}, \frac{- 4}{3})$

$a_{1} a_{2} + b_{1} b_{2} = 4 > 0$

Angle bisectors are $\frac{x - 2 y - 4}{\sqrt{5}} = \pm (\frac{2 x - y - 4}{\sqrt{5}})$

Acute angle bisector is $3 x - 3 y - 8 = 0$

Obtuse angle bisector is $x + y = 0$

And equation of the bank is $(y + \frac{4}{3}) = - 1 (x - \frac{4}{3})$

$\Rightarrow x + y = 0$

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