The equations of the lines through the point (3, 2) which makes an angle of 45⁰ with the line x - 2y = 3 are

1) 3x – y = 7 and x + 3y = 9

2) x – 3y = 7 and 3x + y = 9

3) x – y = 3 and x + y = 2

4) 2x + y = 7 and x – 2y = 9

Solution:

Given line is x – 2y = 3

=> y = ½ x – 3/2

Here slope, m₁ = ½

The required line makes an angle 45⁰ with this line.

So tan 45 = |(m₁ – m₂)/(1 + m₁m₂)|

1 = |(1/2 – m₂)/(1 + ½ m₂)|

2 + m₂ = ±(1 – 2m₂)

Solving we get m₂ = 3 or -1/3

Case 1: when m₂ = 3

Equation of line passing through (3,2) and slope 3 is

(y – y₁) = m(x – x₁)

(y – 2) = 3(x – 3)

y – 3x + 7 = 0

=> 3x – y = 7

Case 2: when m₂ = -1/3

Equation of line passing through (3,2) and slope -1/3 is

(y – y₁) = m(x – x₁)

(y – 2) = -1/3(x – 3)

x + 3y = 9

Hence option (1) is the answer.