Name: Condition for Concurrency of Three Lines
Uploaded: 2019-07-19
Duration: 10 min 26 s
Description: Well, this video beautifully explains the conditions for concurrency of three lines with some interesting examples. Come, let's learn and watch the video...!!

Question

Find the equations of the straight lines, bisecting the angles between the pair of straight lines

2 x + y = 4

and

y + 3 x = 5

, placing first the bisector of the angle in which the origin lies.

Accepted Answer

The equations of the lines are given as,

$2 x + y - 4 = 0$

$y + 3 x - 5 = 0$

Now, making the constant term positive we get,

$- 2 x - y + 4 = 0$

$- 3 x - y + 5 = 0$

The general equation of the angle bisectors to the lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ is given as,

$\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \pm (\frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}})$

Hence, the equation of the given angle bisector can be written as,

$\frac{- 2 x - y + 4}{\sqrt{{(- 2)}^{2} + {(- 1)}^{2}}} = \pm (\frac{- 3 x - y + 5}{\sqrt{{(- 3)}^{2} + {(- 1)}^{2}}})$

$\Rightarrow \frac{- 2 x - y + 4}{\sqrt{5}} = \pm (\frac{- 3 x - y + 5}{\sqrt{10}})$

Now, the origin lies in the bisector, when we consider the positive sign in the above equation. This can be written as,

$\Rightarrow \sqrt{2} (- 2 x - y + 4) = - 3 x - y + 5$

$\Rightarrow (2 \sqrt{2} - 3) x + (\sqrt{2} - 1) y + (5 - 4 \sqrt{2}) = 0$

Considering the negative sign, we get,

$\sqrt{2} (- 2 x - y + 4) = - (- 3 x - y + 5)$

$\Rightarrow (2 \sqrt{2} + 3) x + (\sqrt{2} + 1) y - (5 + 4 \sqrt{2}) = 0$

Hence, the equations of the bisectors are,

$(2 \sqrt{2} - 3) x + (\sqrt{2} - 1) y + (5 - 4 \sqrt{2}) = 0$

$(2 \sqrt{2} + 3) x + (\sqrt{2} + 1) - (5 + 4 \sqrt{2}) = 0$

Find the equations of the straight lines, bisecting the angles between the pair of straight lines $2 x + y = 4$ and $y + 3 x = 5$ , placing first the bisector of the angle in which the origin lies.

Important Questions on The Straight Line, Rectangular Coordinates

Learn and Prepare for any exam you want !

Find the equations of the straight lines, bisecting the angles between the pair of straight lines 2x+y=4 and y+3x=5, placing first the bisector of the angle in which the origin lies.

Important Questions on The Straight Line, Rectangular Coordinates

Learn and Prepare for any exam you want !

Find the equations of the straight lines, bisecting the angles between the pair of straight lines $2 x + y = 4$ and $y + 3 x = 5$ , placing first the bisector of the angle in which the origin lies.