Question

Answer the following question:

Two lines passing through M(2, 3) intersect each other at an angle of 45°. If slope of one line is 2, find the equation of the other line.

Answer 1

Let m be the slope of the other line passing through M(2, 3) and making an angle of 45° with the line whose slope is 2.

∴ tan 45° = `|("m" - 2)/(1 + "m"(2))|`

∴ 1 = `|("m" - 2)/(1 + 2"m")|`

∴ `("m" - 2)/(1 + 2"m")` = ± 1

∴  `("m" - 2)/(1 + 2"m") = 1 or  ("m" - 2)/(1 + 2"m")` = – 1

∴ m – 2 = 1 + 2m or m – 2 = – 1 – 2m

∴ m = – 3 or 3m = 1

∴ m = – 3 or m = `1/3`

When m = – 3, equation of the line is

y – 3 = – 3(x – 2)

∴ y – 3 = – 3x + 6

∴ 3x + y = 9

When m = `1/3`, equation of the line is

y – 3 = `1/3(x - 2)`

∴ 3y – 9 = x – 2

∴ x – 3y + 7 = 0

Hence, equations of required lines are

3x + y = 9 and x – 3y + 7 = 0.