Name: Homogeneous Equation of Second Degree
Uploaded: 2019-07-19
Duration: 5 min 45 s
Description: The second-degree homogeneous equation represents a pair of lines. Watch this video and learn how we can find the equations of two lines from homogeneous equ...

Question

Find the combined equation of lines passing through the origin each of which making an angle of

30 °

with the line

3 x + 2 y - 11 = 0

Accepted Answer

Let the equation of $A B$ be $3 x + 2 y - 11 = 0$

Let, $O A, O B$ be the lines which are making angles of $30 °$ with $A B$ and passing through the origin.

Let slope of $O A$ be $m$ , then equation of $O A$ is $y - 0 = m (x - 0)$

$\Rightarrow m x - y = 0$

Given line is $3 x + 2 y - 11 = 0$

$\Rightarrow y = - \frac{3}{2} x + \frac{11}{2}$

Slope of given line is $m_{1} = \frac{- 3}{2}$ .

Let $m$ be the slope of the two lines making an angle of $30 °$ with the line $3 x + 2 y - 11 = 0$ .

Since the angle between the lines with slopes $m$ and $m_{1}$ is $30 °$ .

So, $\tan θ = |\frac{m - m_{1}}{1 + m \cdot m_{1}}|$

$\Rightarrow \tan 30 ° = |\frac{m - (\frac{- 3}{2})}{1 + m (\frac{- 3}{2})}|$

$\Rightarrow \frac{1}{\sqrt{3}} = |\frac{2 m + 3}{2 - 3 m}|$

Take square on both sides,

$\frac{1}{3} = \frac{{(2 m + 3)}^{2}}{(2 - 3 m)^{2}}$

$\Rightarrow 4 + 9 m^{2} - 12 m = 3 (4 m^{2} + 9 + 12 m)$

$\Rightarrow 4 + 9 m^{2} - 12 m = 12 m^{2} + 27 + 36 m$

$\Rightarrow 3 m^{2} + 48 m + 23 = 0$ .

Therefore, $3 m^{2} + 48 m + 23 = 0$ is one of the auxiliary equation of the two lines.

Substitute $m = \frac{y}{x}$ in the above equation,

$\Rightarrow 3 {(\frac{y}{x})}^{2} + 48 (\frac{y}{x}) + 23 = 0$

$\Rightarrow 3 y^{2} + 48 x y + 23 x^{2} = 0$

$\Rightarrow 23 x^{2} + 48 x y + 3 y^{2} = 0$ is the required pair of straight lines.

Find the combined equation of lines passing through the origin each of which making an angle of $30 °$ with the line $3 x + 2 y - 11 = 0$

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x+2y-11=0

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Find the combined equation of lines passing through the origin each of which making an angle of $30 °$ with the line $3 x + 2 y - 11 = 0$