Question

The angle between the line joining the points (1, -2), (3, 2) and the line x + 2y – 7 = 0 is

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Answer 1

☆ Understanding the concept:-

• To find the angle between the two lines, we have to find the slope of two lines and then use the formula of tan to find the angle between them.

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☆ Formula used :-

$\tt \: 1. \: Let \: us \: consider \: two \: lines \: having \: slope \: \\ \tt \:  m_1 \: and \: m_2 \: and \: let \: \theta \: be \: the \: angle \: between \: \\ \tt \:  two \: lines, \: then \: angle \: \theta \: is \: given \: by \: \: \: \: \: \: \: \: \: \\ \tt \:  ⟼tan \theta \: = \bigg |\dfrac{m_1 \: - m_2}{1 + m_1 \times m_2} \bigg| \: \: \: \: \: \: \: \: \:$

$\tt \:  2. \: Let \: us \: consider \: a \: line \: segment \: joining \: the \\ \tt \:  points \: A(x_1, y_1) \: and \: B(x_2, y_2), \: then \: slope \: \\ \tt \:  of \: line \: joining \: AB \: is \: given \: by \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \tt \:  ⟼ \: m \: = \: \dfrac{y_2 - y_1}{x_2 - x_1} \: \: \: \: \: \: \: \: \: \: \: \: </p><p>$

$\tt \:  3. \: If \: the \: equation \: of \: line \: is \: ax + by + c = 0, \: \\ \tt \:  then \: slope \: is \: given \: by \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \tt \:  ⟼ \: m \: = \: - \dfrac{coefficient \: of \: x}{coefficient \: of \: y} = - \dfrac{a}{b} \: \: \: \: \: \: \: \: \: </p><p>$

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$\large\underline{\bold{Solution :- }}$

$\large\underline{\bold{❥︎Step :- 1 }}$

$\tt \:  ⟼slope \: of \: the \: line \: joining \: the \: \\ \tt \:  points \: (1, -2), \: (3, 2) \: is \: given \: by \: \\ \tt \:  ⟼ \: m_1 \: = \dfrac{2 - ( - 2)}{3 - 1} = \dfrac{4}{2} = 2$

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$\large\underline{\bold{❥︎Step :- 2 }}$

$\tt \:  ⟼ \: slope \: of \: line \: x + 2y - 7 = 0 \: is \: given \: by \\ \tt \:  ⟼ \: m_2 \: = - \dfrac{1}{2}$

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$\large\underline{\bold{❥︎Step :- 3 }}$

$\tt \:  ⟼ \: Let \: \theta \: be \: the \: angle \: be \: the \: lines \: then \: \\ \tt \:  ⟼ \: tan \theta \: = \bigg|\dfrac{2 + \dfrac{1}{2} }{1 + 2 \times \bigg( - \dfrac{1}{2} \bigg)} \bigg| \\ \tt \:  ⟼ \: tan \theta \: = |\dfrac{ \frac{5}{2} }{1 - 1} | \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \tt \:  ⟼ \: tan \theta \: = \infty \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \tt\implies \: \theta \: = \dfrac{\pi}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$