Question

1. Find the angle between the following straight lines.
y=-√3x+5,y=1/√3x-3/√3​

Answer 1

Answer:

Example 15 Find the angle between the lines y − √3x − 5 = 0 and √3y − x + 6 ... Example 15 - Chapter 10 Class 11 Straight Lines. Last updated at Feb. 3, 2020 by Teachoo.

Answer 2

Correct Question:

Find the angle between the following straight lines:

$\displaystyle{\sf\:y\:=\:\sqrt{3}\:x\:+\:5\:\quad\:\&}$

$\displaystyle{\sf\:y\:=\:\dfrac{1}{\sqrt{3}}\:x\:-\:\dfrac{3}{\sqrt{3}}}$

Answer:

The acute angle between the straight lines is 30°.

The obtuse angle between the straight lines is 150°.

Step-by-step-explanation:

We have given the equations of two straight lines in slope intercept form.

We have to find the angle between the two straight lines.

The given equation of the first line is

$\displaystyle{\sf\:y\:=\:\sqrt{3}\:x\:+\:5}$

By comparing with the general equation of a straight line in slope intercept form, we get,

$\displaystyle{\sf\:y\:=\:m\:x\:+\:b}$

$\displaystyle{\therefore\:\sf\:m_1\:=\:\sqrt{3}}$

The given equation of the second straight line is

$\displaystyle{\sf\:y\:=\:\dfrac{1}{\sqrt{3}}\:x\:-\:\dfrac{3}{\sqrt{3}}}$

By comparing with the general equation of a straight line in slope intercept form, we get,

$\displaystyle{\sf\:y\:=\:m\:x\:+\:b}$

$\displaystyle{\therefore\:\sf\:m_2\:=\:\dfrac{1}{\sqrt{3}}}$

Now, we know that,

$\displaystyle{\pink{\sf\:\tan\:\theta\:=\:\left|\:\dfrac{m_1\:-\:m_2}{1\:+\:m_1\:m_2}\:\right|}}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\left|\:\dfrac{\sqrt{3}\:-\:\dfrac{1}{\sqrt{3}}}{1\:+\:\cancel{\sqrt{3}}\:\times\:\dfrac{1}{\cancel{\sqrt{3}}}}\:\right|}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\left|\:\dfrac{\dfrac{\sqrt{3}\:\times\:\sqrt{3}\:-\:1}{\sqrt{3}}}{1\:+\:1}\:\right|}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\left|\:\dfrac{\dfrac{3\:-\:1}{\sqrt{3}}}{2}\:\right|}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\left|\:\dfrac{\dfrac{2}{\sqrt{3}}}{2}\:\right|}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\left|\:\dfrac{\cancel{2}}{\sqrt{3}}\:\times\:\dfrac{1}{\cancel{2}}\:\right|}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\left|\:\dfrac{1}{\sqrt{3}}\:\right|}$

$\displaystyle{\implies\sf\:\tan\:\theta\:=\:\dfrac{1}{\sqrt{3}}}$

$\displaystyle{\implies\sf\:\theta\:=\:30^{\circ}\:\quad\:-\:-\:-\:[\:From\:trigonometric\:table\:]}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\theta\:=\:30^{\circ}}}}}$

∴ The acute angle between the straight lines is 30°.

Now,

Obtuse angle between the straight lines = 180° - Acute angle between the lines

⇒ Obtuse angle between the straight lines = 180° - 30°

⇒ Obtuse angle between the straight lines = 150°

∴ The obtuse angle between the straight lines is 150°.