Find the angle between given lines:y – √3x = 2 and \(\frac{1}{{

1.	Find the angle between given lines:y – √3x = 2 and \(\frac{1}{{\surd 3}}\)x – y = –4 1. 0° 2. 30° 3. 45° 4. 60° 5. None of the above/ More than one of the above.
Answer» Correct Answer - Option 2 : 30° Given: Two lines are, y – √3x = 2 and \(\frac{1}{{\surd 3}}\)x – y = –4 Formula Used: y = mx + c, where, m is the slope of a line. tanθ = \|\(\frac{{m1 - m2}}{{1 + \left( {m1 \times m2} \right)}}\)\| where, m₁ is the slope of the first line. m₂ is the slope of the second line. Calculation: For 1^st equation, ⇒ y – √3x = 2 ⇒ y = 2 + √3x ⇒ m₁ = √3 For 2^nd equation, ⇒ \(\frac{1}{{\surd 3}}\)x – y = –4 ⇒ –y = –4 – \(\frac{1}{{\surd 3}}\)x ⇒ y = 4 + \(\frac{1}{{\surd 3}}\)x ⇒ m₂ = 1/√3 tanθ = \|\(\frac{{m1 - m2}}{{1 + \left( {m1 \times m2} \right)}}\)\| ⇒ tanθ = \|\(\frac{{√ 3 \; - \;1/\surd 3}}{{1 + \left( {√ 3 \times 1/\surd 3} \right)}}\)\| ⇒ tanθ = \|(2/√3)/2 \| ⇒ tanθ = \|1/√3 \| ⇒ tanθ = 1/√3 We know that, tan 30° = 1/√3 ⇒ θ = 30° ∴ Angle between two lines is 30°.

Find the angle between given lines:y – √3x = 2 and \(\frac{1}{{\surd 3}}\)x – y = –4 1. 0° 2. 30° 3. 45° 4. 60° 5. None of the above/ More than one of the above.

Answer» Correct Answer - Option 2 : 30°

Given:

Two lines are,

y – √3x = 2 and \(\frac{1}{{\surd 3}}\)x – y = –4

Formula Used:

y = mx + c,

where,

m is the slope of a line.

tanθ = |\(\frac{{m1 - m2}}{{1 + \left( {m1 \times m2} \right)}}\)|

where,

m₁ is the slope of the first line.

m₂ is the slope of the second line.

Calculation:

For 1^st equation,

⇒ y – √3x = 2

⇒ y = 2 + √3x

⇒ m₁ = √3

For 2^nd equation,

⇒ \(\frac{1}{{\surd 3}}\)x – y = –4

⇒ –y = –4 – \(\frac{1}{{\surd 3}}\)x

⇒ y = 4 + \(\frac{1}{{\surd 3}}\)x

⇒ m₂ = 1/√3

tanθ = |\(\frac{{m1 - m2}}{{1 + \left( {m1 \times m2} \right)}}\)|

⇒ tanθ = |\(\frac{{√ 3 \; - \;1/\surd 3}}{{1 + \left( {√ 3 \times 1/\surd 3} \right)}}\)|

⇒ tanθ = |(2/√3)/2 |

⇒ tanθ = |1/√3 |

⇒ tanθ = 1/√3

We know that, tan 30° = 1/√3

⇒ θ = 30°

∴ Angle between two lines is 30°.