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  • The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3 , find the slopes of the lines

Q : 11        The slope of a line is double of the slope of another line. If tangent of the angle between them is    \frac{1}{3},  find the slopes of the lines

Answers (1)

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Let m_1 \ and \ m_2  are the slopes of lines and \theta is the angle between them
Then, we know that
\tan \theta = \left | \frac{m_2-m_1}{1+m_1m_2} \right |
It is given that    m_2 = 2m_1    and

    \tan \theta = \frac{1}{3}
Now,
\frac{1}{3}= \left | \frac{2m_1-m_1}{1+m_1.2m_1} \right |
\frac{1}{3}= \left | \frac{m_1}{1+2m^2_1} \right |
Now,
3|m_1|= 1+2|m^2_1|\\ 2|m^2_1|-3|m_1|+ 1 = 0\\ 2|m^2_1|-2|m_1|-|m_1|+1=0\\ (2|m_1|-1)(|m_1|-1)= 0\\ |m_1|= \frac{1}{2} \ \ \ \ \ or \ \ \ \ \ \ |m_1| = 1
 Now,
m_1 = \frac{1}{2} \ or \ \frac{-1}{2} \ or \ 1 \ or \ -1
According to which value of m_2 = 1 \ or \ -1 \ or \ 2 \ or \ -2
Therefore, m_1,m_2 = \frac{1}{2},1 \ or \ \frac{-1}{2},-1 \ or \ 1,2 \ or \ -1,-2

Posted by

Gautam harsolia

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