O-Level Coordinate geometry in two dimensions: worked solution

Question

Two points are P(a, 4) and Q(6, −2). Express the perpendicular bisector of PQ in terms of a.

A perpendicular bisector does two things: it passes through the midpoint of the segment, and it is perpendicular to it. So find the midpoint, then the gradient of PQ, then the perpendicular gradient. Step 1, midpoint M of P(a, 4) and Q(6, −2). The midpoint is the average of the coordinates: M = ((a + 6)/2, (4 + (−2))/2) M = ((a + 6)/2, 1) Step 2, gradient of PQ, using gradient = (y₂ − y₁)/(x₂ − x₁) with P(a, 4) and Q(6, −2): m(PQ) = (−2 − 4)/(6 − a) = −6/(6 − a) Multiplying the top and bottom by −1 tidies this to m(PQ) = 6/(a − 6) Step 3, gradient of the bisector. For perpendicular lines the gradients multiply to −1, so m × m(PQ) = −1: m = −1 ÷ [6/(a − 6)] = −(a − 6)/6 = (6 − a)/6 Step 4, the equation. Use y − y₁ = m(x − x₁) through the midpoint M: y − 1 = [(6 − a)/6][x − (a + 6)/2] This is the perpendicular bisector of PQ in terms of a. (The working needs a ≠ 6; if a = 6 then P and Q are vertically in line, PQ has no gradient, and the bisector is simply the horizontal line y = 1.)