Question

Find the acute angle between the pair of lines 12x-4y=5and 4x+2y =7

Answer 1

Answer:

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Step-by-step explanation:

$so \: here \: given \: lines \: are \: \\ 12x - 4y = 5 \: \: and \: \: 4x + 2y = 7 \\ \\ comparing \: first \: equation \: with \\ ax + by = c \\ we \: get \\ \\ a = 12 \: , \: b = - 4 \: ,c = 5 \\ \\ we \: know \: that \\ \\ slope = \frac{ - a}{b} \\ \\ = \frac{ - 12}{ - 4} \\ \\ = \frac{12}{4} \\ \\ m1 = 3$

$similarly \: then\\comparing \: second \: equation \: \: with \: \\ ax + by = c \\ \\ a = 4 \: , \: b = 2 \: , \: c = 7 \\ \\ so \: again \\ \\ m2 = \frac{ - a}{b} \\ \\ = \frac{ - 4}{2} \\ \\ m2= - 2$

$so \: we \: know \: that \\ \\ tan \: θ = | \: \frac{m1 - m2}{1 + m1 \times m2} \: | \\ \\ = | \: \frac{3 - ( - 2)}{1 + 3 \times ( - 2)} \: | \\ \\ = | \: \frac{3 + 2}{1 + ( - 6)} \: | \\ \\ = | \: \frac{5}{1 - 6} \: | \\ \\ = | \: \frac{5}{( - 5)} \: | \\ \\ = | \: - 1 \: | \\ \\ tan \:θ = 1 \\ \\ we \: know \: that \\ tan \: 45 \: \: ie \: \: tan \: \frac{\pi}{4} = 1 \\ \\ hence \: then \\ \\ θ = 45 = (\frac{\pi}{4} ) {}^{c}$