Find the acute angle between the lines 2x + y = 4 and -3x + y + 7 = 0.
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Correct Answer: Option C
Explanation:
\(2x + y = 4 \equiv y = 4 - 2x \implies m_{1} = -2\)
\(-3x + y + 7 = 0 \equiv y = -7 + 3x \implies m_{2} = 3\)
\(\tan \theta = \frac{m_{1} - m_{2}}{1 - m_{1}m_{2}} = \frac{-2 - 3}{1 - (-2)(3)} = \frac{-5}{-5} = 1\)
\(\tan \theta = 1 \implies \theta = 45°\)
\(2x + y = 4 \equiv y = 4 - 2x \implies m_{1} = -2\)
\(-3x + y + 7 = 0 \equiv y = -7 + 3x \implies m_{2} = 3\)
\(\tan \theta = \frac{m_{1} - m_{2}}{1 - m_{1}m_{2}} = \frac{-2 - 3}{1 - (-2)(3)} = \frac{-5}{-5} = 1\)
\(\tan \theta = 1 \implies \theta = 45°\)