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

1.	Find the equation of the line through the point (3, 2) which makes an angle of 45º with the line x – 2y = 3 ?
Answer» Given line: x – 2y = 3 ⇒ y = \(\frac{x}{2}\) - \(\frac{3}{2}\) ....(i) ∴ Its slope = m₁= \(\frac{1}{2}\) Let m₂ be the slope of line through (3, 2). Since this line is inclined at 45º to line (i), tan 45º = \(\bigg\|\frac{m_1-m_2}{1+m_1m_2}\bigg\|\) = \(\bigg\|\frac{\frac{1}{2}-m_2}{1+\frac{1}{2}m_2}\bigg\|\) ⇒\(\frac{\frac{1}{2}-m_2}{1+\frac{1}{2}m_2}\) = ± 1 ⇒ \(\frac{1-2m_2}{2+m_2}=±\,1\) ⇒ 1 – 2m₂ = 2 + m₂ or 1 – 2m₂ = –2 – m₂ ⇒ 3m₂ = –1 or –m₂ = –3 ⇒ m₂ = \(-\frac{1}{3}\) or m₂ = 3 Since the line passes through (3, 2), the equation of the required line is (y – 2) = \(-\frac{1}{3}\)(x – 3) or (y – 2) = 3 (x –3) ⇒ 3y – 6 = – x + 3 or y – 2 = 3x – 9 ⇒ 3y + x – 9 = 0 or y – 3x + 7 = 0.

Find the equation of the line through the point (3, 2) which makes an angle of 45º with the line x – 2y = 3 ?

Answer»

Given line: x – 2y = 3 ⇒ y = \(\frac{x}{2}\) - \(\frac{3}{2}\) ....(i)

∴ Its slope = m₁= \(\frac{1}{2}\)

Let m₂ be the slope of line through (3, 2). Since this line is inclined at 45º to line (i),

tan 45º = \(\bigg|\frac{m_1-m_2}{1+m_1m_2}\bigg|\) = \(\bigg|\frac{\frac{1}{2}-m_2}{1+\frac{1}{2}m_2}\bigg|\)

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

⇒ 1 – 2m₂ = 2 + m₂ or 1 – 2m₂ = –2 – m₂

⇒ 3m₂ = –1 or –m₂ = –3

⇒ m₂ = \(-\frac{1}{3}\) or m₂ = 3

Since the line passes through (3, 2), the equation of the required line is

(y – 2) = \(-\frac{1}{3}\)(x – 3) or (y – 2) = 3 (x –3)

⇒ 3y – 6 = – x + 3 or y – 2 = 3x – 9

⇒ 3y + x – 9 = 0 or y – 3x + 7 = 0.

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