If the lines $y = 3 x + 1$ and $2 y = x + 3$ are equally inclined to the line $y = m x + 4$ . Find the values of m.

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Solution

Let $θ$ be the acute angle which the line $y = m x + 4$ makes with the lines $y = 3 x + 1$ and $2 y = x + 3$ . Then,
$tan θ = | \frac{m - 3}{1 + 3 m} |$

$tan θ = | \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} |$

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

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

$m^{2} - m - 6 = \pm (6 m^{2} - m - 1)$
$5 m^{2} + 5 = 0$ or $7 m^{2} - 2 m - 7 = 0$

$7 m^{2} - 2 m - 7 = 0 ⟹ m = \frac{1 \pm 5 \sqrt{2}}{7}$

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