Question

find the acute angle between the lines
2x - y + 13 \: \: and \: \: \: 2x - 6y + 72x−y+13and2x−6y+7
​

Answer 1

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° .

Answer 2

Answer:

Let the slope of the lines m

1

=2 and m

2

=

3

1

And

tanθ=∣

1+m

1

m

2

m

1

−m

2

∣

=

∣

∣

∣

∣

∣

∣

∣

∣

1+

3

1

.2

3

1

−2

∣

∣

∣

∣

∣

∣

∣

∣

=1

θ=45

∘

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Step-by-step explanation:

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