• Jun 10th 2012, 11:06 PM
mackyme
Find the line passing through the given point and making the given angle with the given line. (2, 1), π/4, 2x – 3y + 2 = 0
• Jun 10th 2012, 11:14 PM
Prove It
Quote:

Originally Posted by mackyme
Find the line passing through the given point and making the given angle with the given line. (2, 1), π/4, 2x – 3y + 2 = 0

The first line can be written as \displaystyle \begin{align*} y = \frac{2}{3}x + \frac{2}{3} \end{align*}, which has a gradient of \displaystyle \begin{align*} \frac{2}{3} \end{align*}.

You should know that the gradient \displaystyle \begin{align*} m_1 \end{align*} makes the line make an angle \displaystyle \begin{align*} \theta_1 \end{align*}with the positive x axis, such that \displaystyle \begin{align*} \tan{\theta_1} = m \end{align*}. So from this line we can see

\displaystyle \begin{align*} \tan{\theta_1} &= \frac{2}{3} \\ \theta_1 &= \tan^{-1}\left(\frac{2}{3}\right) \\ \theta_1 &\approx 0.588 \end{align*}

You should also know that if you have two lines and know the angles they make with the positive x axis, then the angle between them \displaystyle \begin{align*} \theta \end{align*} can be evaluated by \displaystyle \begin{align*} \theta = \theta_2 - \theta_1 \end{align*}.

You have \displaystyle \begin{align*} \theta \end{align*}, you have \displaystyle \begin{align*} \theta_1 \end{align*}. Can you evaluate \displaystyle \begin{align*} \theta_2 \end{align*} and use this to find the gradient of the line you are trying to find the equation of? From there can you find the equation of the line?