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

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Solution

Let $θ$ be the angle which the line $y = m x + 4$ makes with the lines $y = 3 x + 1$ and $2 y = x + 3$ . Then

$∴ t a n θ = ∣ ∣ \frac{m - 3}{1 + 3 m} ∣ ∣$

and $t a n θ = ∣ ∣ ∣ \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} ∣ ∣ ∣ = ∣ ∣ \frac{2 m - 1}{2 + m} ∣ ∣$

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

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

$\Rightarrow m^{2} - m - 6 = \pm (6 m^{2} - m - 1)$

$\Rightarrow 5 m^{2} + 5 = 0 o r 7 m^{2} - 2 m - 7 = 0$

$\Rightarrow m = \frac{1 \pm 5 \sqrt{2}}{7}$

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a

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