First slide

Equation of line in Straight lines

Question

The equation of the bisector of that angle between the lines $x + y = 3$ and $2 x - y = 2$ which contains the point (1, 1) is

Moderate

A

$(\sqrt{5} - 2 \sqrt{2}) x + (\sqrt{5} + \sqrt{2}) y - 3 \sqrt{5} + 2 \sqrt{2} = 0$
B

$(\sqrt{5} + 2 \sqrt{2}) x + (\sqrt{5} - \sqrt{2}) y - 3 \sqrt{5} - 2 \sqrt{2} = 0$
C

$3 x = 10$
D

none of these

Solution

First we re-write the equations of the two lines in such a way that the values of the expressions on the left hand sides of the equality for $x = 1, y = 1$ become positive. Re-writing the given equations, we obtain
$- x - y + 3 = 0 and - 2 x + y + 2 = 0$
Now, we obtain the bisector of the angle containing pointtl., 1) for positive sign. The required bisector is given by
$\begin{array}{l} \frac{- x - y + 3}{\sqrt{(- 1)^{2} + (- 1)^{2}}} = + \frac{- 2 x + y + 2}{\sqrt{(- 2)^{2}} + 1^{2}} \\ (\sqrt{5} - 2 \sqrt{2}) x + (\sqrt{5} + \sqrt{2}) y - 3 \sqrt{5} + 2 \sqrt{2} = 0 . \end{array}$

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