If the angle between two lines is 4 and slope of one of the lines is 1/2, then find the slope of the other line.

Question

If the angle between two lines is $\frac{π}{4}$ and slope of one of the lines is $\frac{1}{2}$ , then find the slope of the other line.

Smart Achievers · Accepted Answer

(1)

$We know that the acute angle θ between two lines$ $with slopes m_{1} and m_{2} is given by$

$\tan θ =$ $| \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}} |$ ...(i)

$Let$ $m_{1} = \frac{1}{2},$ $m_{2} = m$ $and$ $θ = \frac{π}{4}$

$Now, putting these values in (i), we get$

$\tan \frac{π}{4} =$ $| \frac{m - \frac{1}{2}}{1 + \frac{1}{2} m} |$ $\Rightarrow 1 =$ $| \frac{2 m - 1}{2 + m} |$

$which gives$ $\frac{2 m - 1}{2 + m} = 1$ $or$ $\frac{2 m - 1}{2 + m} = - 1$

$Therefore,$ $m = 3$ $or$ $m = - \frac{1}{3}$

$Hence, slope of the other line is$ $3$ $or$ $- \frac{1}{3} .$

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