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

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Let PQ be the line y = 3x + 1 slope of PQ = 3 The angle θ between PQ and QR is

tan Q = \(\big|\frac{m − 3}{1 + 3m}\big|\)

Slope of line PR: 2y = x + 3 is \(\frac{1}{2}\) and slope of QR is m.

The angle θ between PR and QR is

⇒ ∴ m2 − m − 6 = 6m2 − m – 1

2 = − 1

\(\frac{m-3}{1+3m}\) = − \(\frac{2m-1}{m+2}\)

and m2 – m – 6 = – 6m2 + m + 1

7m2 – 2m – 7 = 0

∴ m = \(\frac{1\pm 5\sqrt{2}}{7}\)

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