Find the tangent to the acute angle between the lines 2x + y = 3...

Find the tangent to the acute angle between the lines 2x + y = 3 and 3x - 2y = 5.

Let \(\phi\) be the angle between the two lines.

tan \(\phi\) = \(\frac{m_1 - m_2}{1 + m_1 m_2}\)

where m\(_1\) = slope of line 1; m\(_2\) = slope of line 2.

Line 1: 2x + y = 3 \(\implies\) y = 3 - 2x.

Line 2: 3x - 2y = 5 \(\implies\) -2y = 5 - 3x.

y = \(\frac{3}{2}\)x - \(\frac{5}{2}\).

m\(_1\) = -2, m\(_2\) = \(\frac{3}{2}\).

tan \(\phi\) = \(\frac{-2 - \frac{3}{2}}{1 + (-2 \times \frac{3}{2})}\)

= \(\frac{\frac{-7}{2}}{-2}\)

\(\therefore\) Tan \(\phi\) = \(\frac{7}{4}\).

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