Number and Coordinate Geometry
Number and Coordinate Geometry covers numerical skills - percentages, ratio, scientific notation - and the algebra of the line and the circle on the coordinate plane. The geometry questions reward clear substitution and exact signs.
Number skills
- Percentage change = (new − old)/old × 100%.
- To reverse a percentage increase, divide by (1 + rate); to reverse a decrease, divide by (1 − rate).
- Scientific notation: a × 10^n with 1 ≤ a < 10.
- Ratio: divide a quantity in the ratio a:b by splitting it into a + b parts.
The line
- Slope m = (y₂ − y₁)/(x₂ − x₁).
- Equation through (x₁, y₁): y − y₁ = m(x − x₁); simplify to y = mx + c.
- Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Parallel lines have equal slopes; perpendicular slopes multiply to −1.
The circle
Centre (h, k), radius r: (x − h)² + (y − k)² = r². In expanded form x² + y² + 2gx + 2fy + c = 0, centre (−g, −f), radius √(g² + f² − c).
Worked example
Find the equation of the line through (1, 2) and (4, 8). Slope m = (8 − 2)/(4 − 1) = 6/3 = 2. Using (1, 2): y − 2 = 2(x − 1), so y = 2x. Check with (4, 8): 2(4) = 8.
Traps
- Subtracting coordinates in the wrong order in the slope formula; both differences must follow the same order.
- Using the reciprocal instead of the negative reciprocal for a perpendicular slope.
- Reading the centre of (x − 3)² + (y + 1)² = 25 as (3, 1) instead of (3, −1).
- Forgetting to square the radius in the circle equation.