Question

Answer the following question:

Find the distance of the line 4x − y = 0 from the point P(4, 1) measured along the line making an angle of 135° with the positive X-axis

Answer 1

Let a line L make an angle 135° with positive X-axis.

Required distance = PQ,

where PQ || line L

Slope of PQ = tan135° 

= tan(180° – 45°)

= – tan 45°

= – 1

Equation of PQ is

y – 1 = (– 1)(x – 4)

∴ y – 1 = – x + 4

∴ x + y = 5  ...(i)

To get point Q we solve the equation

4x – y = 0 with (i)

Substituting y = 4x in (i), we get

5x = 5

∴ x = 1

Substituting x = 1 in (i), we get

1 + y = 5

∴ y = 4

∴ Q = (1, 4)

PQ = `sqrt((4 - 1)^2 + (1 - 4)^2`

= `sqrt(9 + 9)`

= `3sqrt(2)`