Co-ordinate Geometry
Straight lines, Intersections, Circles
Why we study this
Coordinate geometry is important because it connects algebra with visual shapes and real-world problems. It helps students plot points, understand lines and curves, find distances and gradients, and solve problems. These skills are useful in maths, science, engineering, and everyday decision-making.
History of Co-ordinate Geometry
René Descartes helped develop coordinate geometry in the 17th century. In his work La Géométrie, published in 1637, he showed how algebra could describe points, lines, and curves on a geometric diagram. By using numbers to locate points, he connected algebra and geometry in a powerful new way. This system became the foundation of the Cartesian coordinate plane used in maths today.
René Descartes was not only a mathematician but also a philosopher who explored how we can know what is true. In his famous phrase, “I think, therefore I am,” he argued that the act of thinking is proof of our existence. His focus on reason, doubt and clear ideas helped shape modern philosophy and connects with coordinate geometry’s use of careful, logical relationships.
Coordinate Geometry Exam Tips
Remember that although this topic requires a lot of algebraic work, you are still doing geometry, and a diagram is usually essential. This does not mean that you need to scale the axes - adding some coordinates of points automatically does this for you.
The advantage of sketching a reasonably accurate diagram is that it gives you a visual check of the answers provided by the algebra.
Straight lines
Equations
General equations of straight lines are y = mx + c (gradient m, y-intercept c), y - k = m(x - h) (gradient m through (h, k)) or ax + by + c = 0. Special cases are x = h (line parallel with the y-axis, and y = k (line parallel with the x-axis). When sketching straight lines, it's usually quickest to find the intercepts with the x- and y- axes (unless it passes through the origin.
Check the gradient looks right (negative is sloping down from left to right, positive if sloping up from left to right).
Parallel and perpendicular lines
Parallel lines have the same gradient.
Perpendicular lines have one gradient which is the negative reciprocal of the other.
It helps to calculate the gradient of a line by rearranging the equation in your head. For example the gradient of 3x + 4y + 1 = 0 is -3/4 as this rearranges to y = -3/4 x - 1/4.
Intersection of curves/lines
Finding points of intersection is simply a matter of solving the equations of the curves/lines simultaneously . Here is an example [text]. Always check your final solutions in both original equations.
Gradient, midpoint, distance
Make sure you know how to calculate gradients, midpoints, and distances.
Finding the centre and the radius from a general circle equation requires completing the square, so make sure you have mastered this. Here is a link to an example. [text]