Coordinate Geometry Lesson
Hello. I am going to teach you everything you need to know about coordinate geometry. I will assume you know nothing — and by the end, you will be plotting points, finding gradients, writing equations of lines and circles, and solving intersections with confidence. Let us begin.
The Coordinate Plane
Coordinate geometry is about describing shapes and lines using numbers. It all starts with the coordinate plane — a flat surface with two perpendicular number lines.
The horizontal line is called the x-axis. The vertical line is called the y-axis. They cross at the origin, which is the point (0, 0).
Every point on the plane is written as (x, y), where x tells you how far to go horizontally and y tells you how far to go vertically.
The Four Quadrants
The x- and y-axes divide the plane into four regions called quadrants:
Quadrant I: x > 0, y > 0 (top-right)
Quadrant II: x < 0, y > 0 (top-left)
Quadrant III: x < 0, y < 0 (bottom-left)
Quadrant IV: x > 0, y < 0 (bottom-right)
A(3, 2): x = 3 > 0, y = 2 > 0 → Quadrant I
B(−1, 4): x = −1 < 0, y = 4 > 0 → Quadrant II
C(−2, −3): x = −2 < 0, y = −3 < 0 → Quadrant III
D(5, −2): x = 5 > 0, y = −2 < 0 → Quadrant IV
Teacher's Tip: Remember the order: x comes before y in the alphabet, and x comes first in (x, y). The x-axis is horizontal — think of the horizon.
Distance Between Two Points & Midpoint
Distance Formula
If we have two points A(x₁, y₁) and B(x₂, y₂), the distance between them is found using Pythagoras' theorem. Imagine drawing a right-angled triangle between the points — the horizontal difference is (x₂ − x₁) and the vertical difference is (y₂ − y₁).
Let A(1, 2) = (x₁, y₁) and B(4, 6) = (x₂, y₂).
x₂ − x₁ = 4 − 1 = 3
y₂ − y₁ = 6 − 2 = 4
d = √(3² + 4²) = √(9 + 16) = √25 = 5
Midpoint
The midpoint of two points is simply the average of their x-coordinates and the average of their y-coordinates.
x-coordinate: (2 + 8)/2 = 10/2 = 5
y-coordinate: (−3 + 5)/2 = 2/2 = 1
Teacher's Tip: For the distance formula, always subtract in the same order — x₂ − x₁ and y₂ − y₁. The squaring gets rid of any negative signs anyway, so it does not matter which point you call (x₁, y₁).
Gradient (Slope)
The gradient (also called slope) measures how steep a line is. It tells you how much y changes when x increases by 1.
The gradient is often written as m and is calculated by dividing the vertical change (rise) by the horizontal change (run).
Types of Gradients
🔵 Positive gradient (m > 0): The line slopes upward from left to right.
🔴 Negative gradient (m < 0): The line slopes downward from left to right.
🟢 Zero gradient (m = 0): The line is horizontal.
⚪ Undefined gradient: The line is vertical (x₂ − x₁ = 0, division by zero).
m = (10 − 2) / (5 − 1) = 8 / 4 = 2
The gradient is positive, so the line slopes upward.
m = (2 − 8) / (7 − 3) = (−6) / 4 = −1.5
The gradient is negative, so the line slopes downward.
Teacher's Tip: Gradient = rise over run. A steep line has a large gradient value; a shallow line has a small gradient value. A horizontal line (sleeping) has gradient 0. A vertical line (standing up) has an undefined gradient.
Equation of a Straight Line
Every straight line has an equation that describes the relationship between x and y for every point on that line. There are three common forms:
Here m is the gradient and c is the y-intercept (where the line crosses the y-axis).
Use this when you know the gradient m and a point (x₁, y₁) on the line.
All terms on one side, equal to zero. This form is tidy and works for all lines including vertical ones.
Finding the Equation From Two Points
Given two points, first find the gradient, then use the point-gradient form with either point.
Step 1: Find the gradient.
m = (11 − 3) / (6 − 2) = 8 / 4 = 2
Step 2: Use y − y₁ = m(x − x₁) with A(2, 3).
y − 3 = 2(x − 2)
y − 3 = 2x − 4
y = 2x − 1
Step 1: m = (−4 − 4) / (3 − (−1)) = (−8) / 4 = −2
Step 2: y − 4 = −2(x − (−1))
y − 4 = −2(x + 1)
y − 4 = −2x − 2
y = −2x + 2
Teacher's Tip: When writing the final answer, give it as y = mx + c whenever possible — it is the easiest form to read and to graph. Always simplify fractions.
Parallel and Perpendicular Lines
Parallel Lines
Two lines are parallel if they have the same gradient. They never meet — they run alongside each other forever.
Line 1: gradient m₁ = 3
Line 2: gradient m₂ = 3
Since m₁ = m₂, the lines are parallel.
Perpendicular Lines
Two lines are perpendicular if they meet at a right angle (90°). The product of their gradients is −1.
In words: the gradient of a perpendicular line is the negative reciprocal.
m₁ = 2
m₂ = −1 / m₁ = −1 / 2 = −0.5
Step 1: gradient of given line m₁ = 3
Step 2: gradient of perpendicular line m₂ = −1/3
Step 3: Use y − y₁ = m₂(x − x₁) with (3, 1):
y − 1 = (−1/3)(x − 3)
y − 1 = (−1/3)x + 1
y = (−1/3)x + 2
Teacher's Tip: To find a perpendicular gradient: flip the fraction and change the sign. 2 → −½, ¾ → −⁴⁄₃, −5 → ⅕. Flip and switch — that is all there is to it.
Equation of a Circle
A circle is defined as all points at a fixed distance (the radius) from a fixed point (the centre). Using the distance formula, we can derive its equation.
The centre of the circle is at (a, b) and the radius is r. If the centre is at the origin (0, 0), the equation simplifies to x² + y² = r².
(x − 3)² means a = 3
(y + 2)² = (y − (−2))² means b = −2
r² = 25 so r = √25 = 5
a = −1, b = 4, r = 3
(x − (−1))² + (y − 4)² = 3²
Step 1: radius r = distance from (2, 1) to (6, 4)
r = √((6 − 2)² + (4 − 1)²) = √(16 + 9) = √25 = 5
Step 2: centre (2, 1), radius 5
(x − 2)² + (y − 1)² = 25
Expanding the Circle Equation
Sometimes the equation is given in expanded form: x² + y² + 2gx + 2fy + c = 0. To find the centre and radius, complete the square:
Centre = (−g, −f), Radius = √(g² + f² − c)
Teacher's Tip: To go from expanded form to centre-radius form, group x terms and y terms separately, then complete the square for each. The sign in the bracket is opposite to what you might expect: (x − a)² means the x-coordinate of the centre is +a.
Circle: (x − 0)² + (y − 0)² = 9 | Line: y = 0x + 0 | Intersections: —
Intersection of Lines and Circles
A line can intersect a circle in three possible ways:
🔵 Two intersections (secant): The line cuts through the circle at two points.
🟢 One intersection (tangent): The line just touches the circle at one point.
🔴 No intersections: The line misses the circle entirely.
How to Find Intersections
Step 1: Substitute the line equation (y = mx + c) into the circle equation.
Step 2: This gives a quadratic in x. Expand and simplify.
Step 3: Use the discriminant Δ = b² − 4ac:
Δ > 0 → two intersection points
Δ = 0 → one intersection point (the line is a tangent)
Δ < 0 → no intersection
Step 4: If intersections exist, solve the quadratic to find the x-coordinates, then substitute back to find the y-coordinates.
Step 1: Substitute y = 2x + 1 into x² + y² = 10:
x² + (2x + 1)² = 10
x² + 4x² + 4x + 1 = 10
5x² + 4x + 1 − 10 = 0
5x² + 4x − 9 = 0
Step 2: Discriminant: Δ = 4² − 4×5×(−9) = 16 + 180 = 196
Δ = 196 > 0 → two intersections
x = (−4 ± √196) / (2×5) = (−4 ± 14) / 10
x = 10/10 = 1 or x = −18/10 = −1.8
Step 3: Find y: when x = 1, y = 2(1) + 1 = 3
When x = −1.8, y = 2(−1.8) + 1 = −3.6 + 1 = −2.6
Tangents to a Circle
A tangent is a line that touches a circle at exactly one point. Key facts:
✅ The radius to the point of tangency is perpendicular to the tangent line.
✅ To find the tangent at a point P on the circle: find the gradient of the radius OP, then the tangent gradient is its negative reciprocal.
Substitute: x² + (2x + 5)² = 5
x² + 4x² + 20x + 25 = 5
5x² + 20x + 20 = 0
Divide by 5: x² + 4x + 4 = 0
Δ = 4² − 4×1×4 = 16 − 16 = 0
Since Δ = 0, the line touches at exactly one point — it is a tangent.
Length of a Tangent From a Point
If a point P lies outside a circle, you can draw two tangents from P to the circle. The length of each tangent from P to the point of contact is:
where d is the distance from P to the centre of the circle and r is the radius.
Circle centre: (2, −3), radius r = 5
Distance d from P(7, 1) to centre (2, −3):
d = √((7 − 2)² + (1 − (−3))²) = √(5² + 4²) = √(25 + 16) = √41
Tangent length = √(d² − r²) = √(41 − 25) = √16 = 4
Teacher's Tip: The discriminant is your best friend for intersection problems. Calculate Δ first to know how many intersection points exist before doing any heavy algebra. If Δ = 0, you have found a tangent!
Practice — Step by Step
Solve each problem one step at a time. Type your answer for each step and click Check Step. You must get each step correct before moving to the next.
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- Gradient from angle: If a line makes angle θ with the positive x-axis, its gradient is m = tan θ. This is useful in mechanics and trigonometry problems.
- Parallel/perpendicular shortcut: For y = mx + c, any line parallel has the same m. Any line perpendicular has m₂ = −1/m₁. Flip it and change the sign.
- Circle equation memory aid: (x − a)² + (y − b)² = r² — the centre coordinates have the opposite signs inside the brackets. If you see (x + 3)², the centre x-coordinate is −3.
- Midpoint check: The midpoint is just the average of coordinates. If your answer is not between the two original points, you made a mistake.
- Distance check: Distance should always be positive. If you get a negative under the square root, check your subtraction.
- Gradient is rise over run: m = (y₂ − y₁)/(x₂ − x₁). Always subtract in the same order — consistent subtraction is critical.
- Equation of a line needs one point and gradient: With just one point and the gradient, you can write the equation using y − y₁ = m(x − x₁). No need to memorise anything else.
- Parallel lines never meet: They have the same gradient but different y-intercepts. If they have the same gradient AND the same intercept, they are the same line.
- Perpendicular lines meet at right angles: The product of their gradients is −1. Vertical and horizontal lines are also perpendicular but use the special case rule.
- Circle centre from expanded form: For x² + y² + 2gx + 2fy + c = 0, the centre is (−g, −f). The signs are flipped. Do not forget.
- Discriminant tells all: In line-circle intersection, compute Δ = b² − 4ac. Δ > 0 → secant, Δ = 0 → tangent, Δ < 0 → miss.
- Mixing up x and y: Remember (x, y) — x comes first, y comes second. A common error is writing (y, x) when plotting points.
- Wrong gradient sign: A line sloping downward has a negative gradient. Students often write positive gradients for downward-sloping lines. Check by sketching.
- Forgetting to flip the sign in circle centre: From (x − 3)² + (y + 2)² = 25, some students say the centre is (−3, 2). It is actually (3, −2). Read the signs carefully.
- Perpendicular gradient error: The perpendicular gradient is the negative reciprocal. A common mistake is just changing the sign without flipping (or flipping without changing the sign).
- Substitution order in distance: You must square the differences first, then add, then square root. Some students add before squaring, giving a wrong answer.
- Forgetting the ± when solving quadratics: When solving x² = k, remember x = ±√k. If you only write x = √k, you miss half the solutions.
The Golden Rule of Coordinate Geometry
Draw a diagram. Every time. A quick sketch will show you whether your gradient is positive or negative, whether two lines look parallel or perpendicular, and whether a line and circle are intersecting. Your eyes will catch mistakes that your algebra misses.
You Have Completed the Coordinate Geometry Lesson
Here is what you have learned:
- ✅ The coordinate plane, quadrants, and plotting points as (x, y)
- ✅ Distance formula: d = √((x₂−x₁)² + (y₂−y₁)²)
- ✅ Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
- ✅ Gradient: m = (y₂−y₁)/(x₂−x₁)
- ✅ Equation of a straight line: y = mx + c and y − y₁ = m(x − x₁)
- ✅ Parallel lines (m₁ = m₂) and Perpendicular lines (m₁ × m₂ = −1)
- ✅ Equation of a circle: (x − a)² + (y − b)² = r²
- ✅ Intersection of lines and circles (solving simultaneously, using Δ)
- ✅ Length of a tangent from a point: PT = √(d² − r²)
Well done. Coordinate geometry is the language of Pure Mathematics — every topic from here on uses these ideas.