Both formulas on one problem

A straight canal crosses a north–south road once.

The rule

The y-axis is the vertical line where across is 0. A point on it has across value 0. Use that in the section formula to find the ratio. Use squared distances to find the point equally far from both ends. Squared distances avoid roots.

Where the axis cuts AB. Take A(−2, 3) and B(6, −1). So x₁ = −2 and x₂ = 6. Set across to 0. So 0 = (6m − 2n)/(m + n). A fraction is 0 when its top is 0. Here the bottom m + n is not 0. So 6m − 2n = 0. That gives 6m = 2n, so n = 3m. The ratio m : n is 1 : 3. Then up is (1×(−1) + 3×3)/4 = 8/4 = 2. The cut is (0, 2).

Equally far from A and B. Let the point be P(0, y). PA² is the distance PA, squared. PA² = 4 + (y − 3)² = y² − 6y + 13. PB² = 36 + (y + 1)² = y² + 2y + 37. Set them equal. The y² terms cancel. That leaves −6y + 13 = 2y + 37. Then −8y = 24, so y = −3.