Question

find the acute angle between the lines
$2x - y + 13 \: \: and \: \: \: 2x - 6y + 7$
​

Answer 1

Answer:

• Acute angle between the lines will be 45°

Step-by-step explanation:

Slope of 1st Line

→ 2x -y +13 = 0

→ y = 2x + 13

therefore, m1 = 2

Slope of 2nd line

→ 2x - 6y +7 = 0

→ y = 2/6 x + 7

→ y = 1/3 x + 7

therefore, m2 = 1/3

$\bigstar\boxed{\bf{tan\theta = \bigg|\frac{m_2-m_1}{1+m_1m_2}\bigg| }}$

$\sf\dashrightarrow\tan\theta = \bigg|\frac{\frac{1}{3} -2 }{1+\frac{1}{3}\times2}\bigg|\\\\\sf\dashrightarrow\tan\theta = \bigg|\frac{\frac{-6+1}{3} }{1+\frac{2}{3}}\bigg|\\\\\sf\dashrightarrow\tan\theta = \bigg|\frac{\frac{-5}{3} }{\frac{3+2}{3}}\bigg|\\\\\sf\dashrightarrow\tan\theta = \bigg|\frac{\frac{-5}{3} }{\frac{5}{3}}\bigg|\\\\\sf\dashrightarrow\tan\theta = \bigg|-1\bigg|\\\\\sf\dashrightarrow\tan\theta = 1 \\\\\sf\dashrightarrow\theta = 45^{\circ}$

Answer 2

Answer:

Given :-

➳ 2x - y + 13

➳ 2x - 6y + 7

To Find :-

➳ What is the acute angle between the lines.

Formula Used :-

$\clubsuit$ Angle between two lines Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{tan\: \theta =\: \bigg| \dfrac{m_2 - m_1}{1 + m_1m_2}\bigg|}}}\: \: \: \bigstar$

Solution :-

Given Equation :

$\mapsto \sf 2x - y + 13 =\: 0$

$\mapsto \sf 2x - 6y + 7 =\: 0$

Let,

$\leadsto \bf Slope\: of\: line\: (1) =\: m_1$

$\leadsto \bf Slope\: of\: line\: (2) =\: m_2$

$\sf\bold{\underline{\purple{\clubsuit\: In\: case\: of\: slope\: (1)\: :-}}}$

Given Equation :

$\longrightarrow \bf 2x - y + 13 =\: 0$

So,

$\implies \sf\bold{\blue{m_1 =\: - \dfrac{a_1}{b_1}}}$

$\implies \sf m_1 =\: - \dfrac{2}{- 1}$

$\implies \sf\bold{\green{m_1 =\: 2}}$

Again,

$\sf\bold{\underline{\purple{\clubsuit\: In\: case\: of\: slope\: (2)\: :-}}}$

Given Equation :

$\longrightarrow \bf 2x - 6x + 7 =\: 0$

So,

$\implies \sf\bold{\blue{m_2 =\: - \dfrac{a_2}{b_2}}}$

$\implies \sf m_2 =\: - \dfrac{\cancel{2}}{- \cancel{6}}$

$\implies \sf m_2 =\: - \dfrac{1}{- 3}$

$\implies \sf\bold{\green{m_2 =\: \dfrac{1}{3}}}$

Now, we have to find the acute angle between the lines :

Given :

• m₁ = 2
• m₂ = 1/3

According to the question by using the formula we get,

$\implies \bf tan\: \theta =\: \bigg| \dfrac{m_2 - m_1}{1 + m_1m_2}\bigg|$

$\implies \sf tan\: \theta =\: \Bigg| \dfrac{\dfrac{1}{3} - 2}{1 + 2 \times \dfrac{1}{3}}\Bigg|$

$\implies \sf tan\: \theta =\: \Bigg| \dfrac{\dfrac{1 - 6}{3}}{1 + \dfrac{2}{3}}\Bigg|$

$\implies \sf tan\: \theta =\: \Bigg| \dfrac{\dfrac{- 5}{3}}{\dfrac{3 + 2}{3}}\Bigg|$

$\implies \sf tan\: \theta =\: \Bigg| \dfrac{\dfrac{- 5}{3}}{\dfrac{5}{3}}\Bigg|$

$\implies \sf tan\: \theta =\: \bigg| \dfrac{- 5}{3} \times \dfrac{3}{5}\bigg|$

$\implies \sf tan\: \theta =\: \bigg| \dfrac{- 15}{15}\bigg|$

$\implies \sf tan\: \theta =\: \bigg| - 1\bigg|$

$\implies \sf tan\: \theta =\: 1$

$\implies \sf\bold{\red{\theta =\: 45^{\circ}}}$

$\therefore$ The acute angle between the lines is 45° .