Question

The slope of the line is double of the slope of another line.if tangents of the angle between them is 1/3, find the slope of the other line. full solved

Answer 1

this is the answer to your question

Answer 2

Answer and explanation:

Given : The slope of the line is double of the slope of another line. If tangents of the angle between them is $\frac{1}{3}$.

To find : The slope of the other line ?

Solution :

Let the slope of one line is $m_1$

The slope of the line is double of the slope of another line.

i.e. the slope of another line is $m_2=2m_1$

If tangents of the angle between them is $\frac{1}{3}$

i.e. $\tan\theta=\frac{1}{3}$

The formula is given by,

$\tan\theta=|\frac{m_2-m_1}{1+m_1m_2}|$

Substitute the values,

$\frac{1}{3}=|\frac{2m_1-m_1}{1+m_1(2m_1)}|$

$\frac{1}{3}=|\frac{m_1}{1+2m_1^2}|$

$\frac{1}{3}=\frac{m_1}{1+2m_1^2}$ ....(1)

or $-\frac{1}{3}=\frac{m_1}{1+2m_1^2}$ ....(2)

Solving (1),

$1+2m_1^2=3m_1$

$2m_1^2-3m_1+1=0$

$2m_1^2-2m_1-m_1+1=0$

$2m_1(m_1-1)-1(m_1-1)=0$

$(2m_1-1)(m_1-1)=0$

$m_1=1,\frac{1}{2}$

Solving (2),

$1+2m_1^2=-3m_1$

$2m_1^2+3m_1+1=0$

$2m_1^2+2m_1+m_1+1=0$

$2m_1(m_1+1)-1(m_1+1)=0$

$(2m_1+1)(m_1+1)=0$

$m_1=-1,-\frac{1}{2}$

Now,

1) When $m_1=1$

$m_2=2(1)=2$

2) When $m_1=\frac{1}{2}$

$m_2=2(\frac{1}{2})=1$

3) When $m_1=-1$

$m_2=2(-1)=-2$

4) When $m_1=-\frac{1}{2}$

$m_2=2(-\frac{1}{2})=-1$