Question

Determine whether the line through points (–2, 3) and (4, 1) is perpendicular to the line 3x = y + 1.

Does line 3x = y + 1 bisect the line segment joining the two given points?

Answer 1

Let A = (−2, 3) and B = (4, 1)

Slope of AB = m₁ = `(1 - 3)/(4 + 2) = (-2)/6 =(-1)/3`

Equation of line AB is

y – y₁ = m₁(x – x₁)

`y - 3 = (-1)/3 (x + 2)`

3y − 9 = −x − 2

x + 3y = 7    ...(1)

Slope of the given line 3x = y + 1 is 3 = m₂.

∴ m₁ × m₂ = −1

Hence, the line through points A and B is perpendicular to the given line.

Given line is 3x = y + 1   ...(2)

Solving (1) and (2), we get,

x = 1 and y = 2

So, the two lines intersect at point P = (1, 2).

The co-ordinates of the mid-point of AB are

 `((-2 + 4)/2, (3 + 1)/2) = (1, 2) = P`

Hence, the line 3x = y + 1 bisects the line segment joining the points A and B.