Question

solve
Find angle between line​

Answer 1

Answer:

$\boxed{ \huge{ \pink{ a) 2 \: {tan}^{ - 1} ( \dfrac{b}{a})}}}$

Step-by-step explanation:

$\bold{ \underline{ \red{To \: find}}} \longrightarrow \\ angle \: between \: lines \\ \dfrac{x}{a} + \dfrac{y}{b} = 1 \\ \dfrac{x}{a} - \dfrac{y}{b} = 1$

$\bold{ \underline {\orange{Concept \: used}}} \longrightarrow \\ 1)slope \: of \: a \: line = - \dfrac{coefficient \: of \: x}{coefficient \:of \: y } \\ 2)angle \: between \: two \: lines \\ tan \theta = | \dfrac{m(1) - m(2)}{1 + m(1)m(2)} | \\ 3)tan2 \theta = \dfrac{2tan \theta}{1 - {tan}^{2} \theta}$

$\bold{ \underline{ \blue{Solution}}} \longrightarrow \\ \dfrac{x}{a} + \dfrac{y}{b} = 1.......(1) \\let \: slope \: of \: line \: (1) \: is \: m(1) \\ m(1) = - \dfrac{coefficient \: of \: x}{coefficient \: of \: y} \\ m(1) = - \dfrac{ \dfrac{1}{a} }{ \dfrac{1}{b} } \\ m(1) = - \frac{b}{a} \\ \dfrac{x}{a} - \dfrac{y}{b} = 1......(2) \\ let \: slope \: of \: line \: (2) \: is \: m(2) \\ m(2) = - \dfrac{ \dfrac{1}{a} }{ - \dfrac{1}{b} } \\ m(2) = \dfrac{b}{a}$

$now$

$let \: angle \: between \: lines \: is \: \theta \\ tan \theta = | \dfrac{m(1) - m(2)}{1 + m(1)m(2)} | \\ tan \theta = | \dfrac{ - \dfrac{b}{a} - \dfrac{b}{a} }{1 + ( - \dfrac{b}{a})( \dfrac{b}{a} )} | \\ tan \theta = | \dfrac{ - \dfrac{2b}{a} }{1 + \dfrac{ {b}^{2} }{ {a}^{2} } } | \\ tan \theta = \dfrac{2 \dfrac{b}{a} }{1 + {( \dfrac{b}{a}) }^{2} } \\ let \: \\\dfrac{b}{a} = tan \alpha = > \alpha = {tan}^{ - 1} ( \dfrac{b}{a} ) \\ tan \theta = \dfrac{2tan \alpha }{1 - {tan}^{2} \alpha } \\ tan \theta = tan2 \alpha \\ \theta = 2 \alpha \\ \theta = 2 \: {tan}^{ - 1} ( \dfrac{b}{a})$