Modelling with Straight Lines (Edexcel A-Level Mathematics): Revision Notes

3.1.4 Modelling with Straight Lines

Straight Lines

Gradient of a Line Segment

Midpoint of a Line Segment

Length of a Line Segment

Example Problem: Finding Coordinates of Points on a Line

Solution:

1. Equation of Line $l$:

• The line has gradient $-2$ and passes through $(3, 5)$. $y - 5 = -2(x - 3)$ $y - 5 = -2x + 6$ $y = -2x + 11$

1. Using Pythagoras' Theorem:

• Let $B$ be $(x, -2x + 11)$.
• Distance $AB = 6\sqrt{5}$: $\sqrt{(6 - 2x)^2 + (x - 3)^2} = 6\sqrt{5}$

2. Solve the Distance Equation: $(6 - 2x)^2 + (x - 3)^2 = 180$ $36 - 24x + 4x^2 + x^2 - 6x + 9 = 180$ $5x^2 - 30x + 45 = 180$ $5x^2 - 30x - 135 = 0$ $x^2 - 6x - 27 = 0$ $(x - 9)(x + 3) = 0$ $x = 9 \quad \text{or} \quad x = -3$
3. Find Corresponding $y$ -Coordinates:

• For $x = 9$: $y = -2(9) + 11 = -18 + 11 = -7$ $\therefore B(9, -7)$
• For $x = -3$: $y = -2(-3) + 11 = 6 + 11 = 17$ $\therefore B(-3, 17)$

Final Answers:

• The coordinates of the possible points $B$ are $(9, -7)$ and $(-3, 17)$.