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A Levels / Maths

Coordinate Geometry & Parametric Equations

Master A-Level coordinate geometry with comprehensive notes on the equation of a straight line, the equation of a circle, and converting parametric equations into Cartesian forms. Learn to apply fundamental circle theorems, calculate gradients, and determine the exact tangent of a circle.

In coordinate geometry, the equation of a straight line can be constructed using the standard formula y−y1=m(x−x1)y - y_1 = m(x - x_1) when you know the gradient and one specific coordinate point.
Another standard mathematical representation is the straight line equation ax+by+c=0ax + by + c = 0, which is particularly useful for expressing lines with integer coefficients instead of fractions.
The gradient of a straight line dictates its steepness, and two parallel lines will always possess identical numerical gradients.
When dealing with perpendicular lines, their gradients are negative reciprocals of one another, mathematically defined by the condition m1×m2=−1m_1 \times m_2 = -1.
The fundamental equation of a circle is written in the standard format (x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2, where the centre of the circle is at (a,b)(a, b) and the radius is rr.
When a circle equation is provided in an expanded polynomial form, you must use the completing the square method to extract the exact coordinates of the centre and the radius length.
Applying circle theorems is essential for coordinate geometry problems; for example, the angle formed inside a semi circle by a triangle will always be exactly a right angle.
A vital rule of circle geometry dictates that a perpendicular line drawn straight from the centre of a circle to a chord will bisect that chord perfectly in half.
A tangent of a circle is a straight line that touches the edge of the circle at exactly one point, and it never crosses into the interior of the circle.
A core mathematical property is that the radius of a circle is always perfectly perpendicular to the tangent line at the exact point where the tangent meets the circumference.
Parametric equations define both the xx and yy parametric coordinates of a curve separately in terms of a third independent mathematical variable, typically denoted as tt or θ\theta.
To convert a curve from parametric form back into a single Cartesian equation, you must algebraically substitute and eliminate the third parameter to link xx and yy directly.
Finding the Cartesian coordinates from parametric equations allows you to plot complex, overlapping curves on a standard Cartesian plane that cannot be expressed as a simple y=f(x)y = f(x) function.
Parametric equations are extensively utilised in real-world mathematical modelling to accurately describe the specific trajectory and changing position of an object moving over time.
Exam questions frequently combine straight line equations and circle formulas, requiring simultaneous equations to calculate the exact intersection points where a straight line crosses a circle.
A-Level Coordinate Geometry & Parametric Equations | TSL