How to Find Gradient, Midpoint Formula, and Solving Line Intersections Made Easy!

Page 2: Lines and Intersections

This page delves into equations of lines and their intersections, building upon the gradient concepts from the previous section.

Definition: The equation of a line can be expressed as y = mx + c (GCSE level) or y-y₁ = m$x-x₁$ (point-slope form).

Example: Finding the equation of a line passing through $-1,4$ and $2,-3$ demonstrates practical application of line equations.

Highlight: Line intersections can be solved either algebraically using simultaneous equations or graphically by plotting.

Vocabulary: Y-intercept refers to the point where a line crosses the y-axis, represented by 'c' in y = mx + c.

Page 3: Circles and Tangents

This page explores circle equations and tangent properties, representing more advanced concepts in coordinate geometry.

Definition: The general equation of a circle with center (a,b) is $x-a$² + $y-b$² = r², where r is the radius.

Example: Finding the center and radius of x² - 2x + y² + 4y - 7 = 0 through completing the square method.

Highlight: Tangents are always perpendicular to the radius at the point of contact with the circle.

Vocabulary: Normal refers to a line perpendicular to a curve at a specific point, often used in context with tangents.

Page 1: Fundamentals of Coordinate Geometry

This page introduces core concepts of coordinate geometry, focusing on gradients and distances between points. The content progresses from basic GCSE concepts to more advanced applications.

Definition: Gradient is calculated using the formula $y₂-y₁$/$x₂-x₁$ where (x₁,y₁) and (x₂,y₂) are coordinates of two points on a line.

Example: Finding the gradient between points A(2,1) and B(4,7) demonstrates practical application of the gradient formula.

Highlight: Parallel lines have identical gradients $M₁ = M₂$, while perpendicular lines have gradients that are negative reciprocals $M₁M₂ = -1$.

Vocabulary: Modulus in coordinate geometry refers to the absolute value, ensuring only positive distances are considered.