Find the equation of the line through the point (3, 2) which make an a

1.	Find the equation of the line through the point (3, 2) which make an angle of 45° with the line x – 2y = 3.
Answer» The given line is x – 2y = 3 ⇒ y = \(\frac{x}{2}\) − \(\frac{3}{2}\) ∴ Slope, m₁ = \(\frac{1}{2}\) Let ‘m’ be the slope of line AB which passes through (3, 2) Since the angle between the two lines is 60°. ∴ tan 45° = ± \(\frac{m_2 − m_1}{1 + m_1m_2}\) ⇒ 1 = ± \(\frac{m_2 − \frac{1}{2}}{1 + \frac{1}{2}m_2}\) ⇒ = ± \(\frac{2m_2 − 1}{ m_2 + 2}\) ∴ \(\frac{2m_2 − 1}{ m_2 + 2}\) = 1 or \(\frac{2m_2 − 1}{ m_2 + 2}\) = −1 ⇒ m₂ = 3 or m₂ = − \(\frac{1}{3}\). ∴ Equation of AB is y – 2 = 3 (x – 3) ⇒ 3x – y = 7 (m₂ = 3) Or, y − 2 = − \(\frac{1}{3}\)(x − 3) (m₂ = − \(\frac{1}{3}\)) ⇒ x + 3y = 9

Find the equation of the line through the point (3, 2) which make an angle of 45° with the line x – 2y = 3.

Answer»

The given line is

x – 2y = 3

⇒ y = \(\frac{x}{2}\) − \(\frac{3}{2}\)

∴ Slope, m₁ = \(\frac{1}{2}\)

Let ‘m’ be the slope of line AB which passes through (3, 2)

Since the angle between the two lines is 60°.

∴ tan 45° = ± \(\frac{m_2 − m_1}{1 + m_1m_2}\)

⇒ 1 = ± \(\frac{m_2 − \frac{1}{2}}{1 + \frac{1}{2}m_2}\)

⇒ = ± \(\frac{2m_2 − 1}{ m_2 + 2}\)

∴ \(\frac{2m_2 − 1}{ m_2 + 2}\) = 1 or \(\frac{2m_2 − 1}{ m_2 + 2}\) = −1

⇒ m₂ = 3 or m₂ = − \(\frac{1}{3}\).

∴ Equation of AB is

y – 2 = 3 (x – 3)

⇒ 3x – y = 7 (m₂ = 3)

Or,

y − 2 = − \(\frac{1}{3}\)(x − 3) (m₂ = − \(\frac{1}{3}\))

⇒ x + 3y = 9