Further Mathematics · Lesson 11

Coordinate geometry: straight lines & circles

WAECNECO

What the syllabus asks

WAEC: Coordinate geometry: distance between two points, mid-point, division of a line segment in a given ratio, gradient, equation of a straight line in various forms, parallel and perpendicular lines, angle between two lines, perpendicular distance from a point to a line, area of a triangle; equation of a circle (centre–radius and general form), tangent and normal to a circle, intersection of line…

NECO: Coordinate geometry: distance between two points, mid-point, division of a line segment in a given ratio, gradient, equation of a straight line in various forms, parallel and perpendicular lines, angle between two lines, perpendicular distance from a point to a line, area of a triangle; equation of a circle (centre–radius and general form), tangent and normal to a circle, intersection of line…

What you will learn

Calculate distance, midpoint and points dividing a segment in a ratio.
Find gradients and equations of straight lines in all common forms.
Apply conditions for parallel and perpendicular lines and find angles between lines.
Find perpendicular distances and areas of triangles from coordinates.
Write and interpret equations of circles; find centre and radius from the general form.
Find equations of tangents and normals, and points where a line meets a circle.

Key points

Distance √[(Δx)² + (Δy)²]; midpoint average of coordinates.
Ratio m : n point: ((nx₁ + mx₂)/(m + n), (ny₁ + my₂)/(m + n)).
Gradient m = Δy/Δx = tan θ; y − y₁ = m(x − x₁).
Parallel m₁ = m₂; perpendicular m₁m₂ = −1; tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.
Distance to a line |ah + bk + c|/√(a² + b²); area ½|x₁(y₂ − y₃) + …|.
Circle (x − a)² + (y − b)² = r²; general form centre (−g, −f), r = √(g² + f² − c).
Tangent ⟂ radius; tangent length √(h² + k² + 2gh + 2fk + c).
Line–circle: substitute; discriminant gives 2, 1 or 0 points.

Start here: Distance and midpoint

Coordinates let us turn geometry into algebra. Think of a map of a town laid on a grid: your house at (1, 2) and the market at (7, 10), in hundreds of metres.

Distance AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The distance formula is just Pythagoras: the horizontal and vertical gaps form the two shorter sides of a right-angled triangle.

Worked example 11.1

Find the distance and midpoint of A(1, 2) and B(7, 10).

Δx = 6, Δy = 8.
AB = √(36 + 64) = √100 = 10.
M = (8/2, 12/2) = (4, 6).

AB = 10 units; M(4, 6)

Worked example 11.2

M(3, −1) is the midpoint of PQ, where P is (−2, 4). Find Q.

(−2 + x)/2 = 3 → x = 8. We reverse the midpoint formula.
(4 + y)/2 = −1 → y = −6.

Q(8, −6)

Try it yourself. Find the distance between (−3, 4) and (2, −8).

Show solution

√(25 + 144) = √169 = 13.

These two formulas also prove geometric facts. For example, to show a triangle is isosceles, calculate the lengths of its three sides and show two are equal; to show a quadrilateral is a parallelogram, show its diagonals have the same midpoint.

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Practice questions

1. Find the distance between (2, −1) and (−4, 7).

Show worked answer

√(36 + 64) = 10.

2. Find the line through (−1, 4) parallel to 3x − y + 2 = 0.

Show worked answer

m = 3: 3x − y + 7 = 0.

3. Find the acute angle between y = 3x and y = −x/3 + 1.

Show worked answer

Product −1: 90°.

4. Distance from (1, −2) to 4x − 3y + 5 = 0.

Show worked answer

|4 + 6 + 5|/5 = 3.

5. Find the centre and radius of x² + y² + 2x − 8y + 8 = 0.

Show worked answer

(−1, 4), r = √(1 + 16 − 8) = 3.

Key words

Cartesian plane: The plane with perpendicular x- and y-axes.
Gradient (slope): Rise over run; measures steepness.
Intercept: Where a line crosses an axis.
Collinear: Lying on the same straight line.
Perpendicular: At right angles.
Locus: The path of points satisfying a condition.
Radius: Distance from centre to any point on a circle.
Tangent: A line touching a curve at exactly one point.
Normal: A line perpendicular to the tangent at the point of contact.
Chord: A line segment joining two points on a circle.
Internal division: A point between A and B dividing AB in a given ratio.
General form of a circle: x² + y² + 2gx + 2fy + c = 0.

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