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i am studying for a test and i seem to of stumbled across a question which i found the answer to, but there seem to be others answers aswell. I am confused and would appreciate any help.

The question: The slope of a line is double of the slope of another line. If Tan. of the angle between them is 1/3, find the slopes of the line.

In order to solve this, i used a formule m2-m1/1+m1m2 = Tan Theta = 1/3

Solving the quadratic equation by taking m2 = 2m and m1 = m,i got 1 or 2, but the answer in my book is different and strange. The answer: "1 and 2, or 1/2 and 1, or -1 and -2, or -1/2 and -1"

If possible, can anybody please help me how that answer was reached?:/

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HINT:

We have $$\left|\frac{m_2-m_1}{1+m_1m_2}\right|=\frac13$$

If $m_2=2m_1,$ we get $$\left|\frac{m_1}{1+2m_1^2}\right|=\frac13$$

$$\implies 1+2m_1^2=3|m_1|$$

Now for real $b,$ $$|b|=\begin{cases} +b &\mbox{if } b\ge0 \\ -b & \mbox{if } b<0 \end{cases} $$

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  • $\begingroup$ Yes i already solved this part, i solved the quadratic equation and ended up with 1 or 2. Problem is, how do i get the answers specified as "1 and 2, or 1/2 and 1, or -1 and -2, or -1/2 and -1". I realize i may be missing something critical, please forgive me if i am :/ $\endgroup$
    – Aayush Agrawal
    Nov 25, 2013 at 6:39
  • $\begingroup$ @aayush, please find the edited version $\endgroup$
    – lab bhattacharjee
    Nov 25, 2013 at 6:41
  • $\begingroup$ Yes, i did. I am sorry but i am still confused. I think i am missing something critical, i only have introductory level knowledge of the topic :/ Can you tell me how the 1+2m1^2=3m1 is solved, or how the real b portion helps? $\endgroup$
    – Aayush Agrawal
    Nov 25, 2013 at 6:43
  • $\begingroup$ @aayush, if real $m_1\ge0$ we find $2m_1^2-3m+1=0\implies m_1=1,\frac12$ and similarly for $m<0$ $\endgroup$
    – lab bhattacharjee
    Nov 25, 2013 at 6:45
  • $\begingroup$ Thank you! I can't believe i didnt see that :/ $\endgroup$
    – Aayush Agrawal
    Nov 25, 2013 at 6:49

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