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

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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:

`tan theta = (m_1-m_2)/(1+m_1m_2) `

Since the angle is arctan (1/2), we know that `tan theta = 1/2 `

Then ` 1/2=(m-2/3)/(1+m*2/3) ` where m is the slope of the required line.

So `1/2+1/3m=m-2/3 `

`2/3 m = 7/6 ==> m=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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