Question

Points A and B have co-ordinates (7, −3) and (1, 9) respectively. Find the equation of the perpendicular bisector of the line segment AB, and the value of ‘p’ of (−2, p) lies on it.

Points A and B have co-ordinates (7, −3) and (1, 9) respectively. Find the equation of the perpendicular bisector of the line segment AB.

Answer 1

Let PQ be the perpendicular bisector of AB intersecting it at M.

Here, using the midpoint formula to calculate the coordinates of M:

`M = ((x_1 + x_2)/2, (y_1 + y_2)/2)`

`M = ((7 + 1)/2, (-3 + 9)/2)`

`M = (8/2, 6/2)`

∴ M = (4, 3)

Now, slope of PQ = `1/2`  ...(m₁ × m₂ = −1)

Using the point slope formula:

y − y₁ = m(x − x₁)

`y - 3 = 1/2 (x - 4)`

2(y − 3) = x − 4

2y − 6 = x − 4

∴ x − 2y + 2 = 0

Since the point (−2, p) lies on the perpendicular bisector x − 2y + 2 = 0,

We substitute x = −2 and y = p into the equation:

−2 − 2p + 2 = 0

−2p = 0

∴ p = 0

Hence, the equation of the perpendicular bisector of the line segment AB is x − 2y + 2 = 0, and the value of ‘p’ which lies on it is p = 0.