Fill in the blanks

Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ____.

Given equation is:-

x – 2y = 3

⇒ x – 3 = 2y

…(i)

Now, we have to find the slope of eq. (i)

Since, the eq. (i) is in y = mx + b form.

So, slope of eq. (i) is

Now, we have to find the equation which is passing through the point (3, 2).

We know that, if a line passing through the point (x₁, y₁) then the equation of line is

y – y₁ = m (x – x₁)

So, here x₁ = 3 and y₁ = 2

∴ y – 2 = m(x – 3) …(ii)

Now, it is given that the angle between the given two lines is 45°.

Putting the values of m₁ and m₂ in above eq., we get

⇒ 2m – 1 = 2 + m or -(2m – 1) = 2 + m

⇒ 2m – m = 2 + 1 or -2m + 1 – m = 2

⇒ m = 3 or -3m = 1

or

Putting the value of m = 3 in eq. (ii), we get

y – 2 = 3(x – 3)

⇒ y – 2 = 3x – 9

⇒ 3x – y – 9 + 2 = 0

⇒ 3x – y – 7 = 0

Putting the value of in eq. (ii), we get

⇒ 3(y – 2) = 3 – x

⇒ 3y – 6 = 3 – x

⇒ x + 3y – 6 – 3 = 0

⇒ x + 3y – 9 = 0