Blog: Find the equation of the line passing through (0,0) making an angle of Arctan 1/2 from the line 3y=2x

Tuesday, July 13, 2010

We want to find the equation of the line passing through the origin that makes an angle of arctan=1/2 with the line 3y=2x:

The slope of the given line is 2/3, and the given line also passes through the origin.

The angle between the lines is related by the following:

$\displaystyle{\tan{\theta}}=\frac{{{m}_{{1}}-{m}_{{2}}}}{{{1}+{m}_{{1}}{m}_{{2}}}}$

Since the angle is arctan (1/2), we know that $\displaystyle{\tan{\theta}}=\frac{{1}}{{2}}$

Then $\displaystyle\frac{{1}}{{2}}=\frac{{{m}-\frac{{2}}{{3}}}}{{{1}+{m}\cdot\frac{{2}}{{3}}}}$ where m is the slope of the required line.

So $\displaystyle\frac{{1}}{{2}}+\frac{{1}}{{3}}{m}={m}-\frac{{2}}{{3}}$

$\displaystyle\frac{{2}}{{3}}{m}=\frac{{7}}{{6}}=\Rightarrow{m}=\frac{{7}}{{4}}$

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The equation is y=7/4x or 4y=7x

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The given line and the required line:

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