Lines and circles in the coordinate plane
MathematicsVectors & Coordinate GeometryAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Vectors & Coordinate Geometry
Line Δ: ax + by + c = 0 has normal vector n = (a; b) and direction vector u = (−b; a) drawn right on the figure; circle (C) has center I and radius R. Drag center I, the free point M, or the whole line: distances d(I, Δ) and d(M, Δ) are computed from |ax₀ + by₀ + c|/√(a² + b²), and the relative position of Δ and (C) (secant, tangent, or no intersection) along with any intersection points, the angle between Δ and Ox, and the angle between Δ and IM update instantly. The Two points screen gives the midpoint, the distance AB from a right triangle, the gradient, the equation of AB, perpendicular gradients, the perpendicular bisector, the point dividing AB in a ratio and lines through a point parallel or perpendicular to AB; the Polygon screen finds the perimeter and the shoelace area from the vertex coordinates. The Line forms screen writes the line d₁ through A(x₀, y₀) with direction vector u = (u₁, u₂) in vector, parametric, continuous, general and slope–intercept form at once while A and u are dragged; a point P = A + t·u moves along d₁ as t changes, and a second line d₂ gives the relative position (intersecting with the angle between them, parallel or coincident) from the direction vectors and the solution of the system.
Lesson: Equation of a line, distance and angle, equation of a circle
What it shows
A line Δ: ax + by + c = 0 has normal vector n = (a; b) and direction vector u = (−b; a). Comparing the distance from the circle's center I to Δ with the radius R shows whether the line cuts, touches or misses the circle. The Two points screen builds the basics from A and B: midpoint, distance by Pythagoras, gradient m = Δy/Δx, the equation of AB in two forms, perpendicular gradients with m₁ × m₂ = −1, the perpendicular bisector and division of AB in a ratio. The Polygon screen finds perimeter and area with the shoelace formula. The Line forms screen links the vector, parametric, continuous, general and slope–intercept equations of one line and decides how two lines meet from their direction vectors.
How to use
On Line & circle, enter Coefficient a, Coefficient b and Coefficient c, set the center I and Radius R, then drag the line Δ until d(I, Δ) equals R. On Two points, drag A and B and tick Right triangle (Δx, Δy), Perpendicular bisector, Point P on AB or Lines through C; the AP : PB sliders move P. On Polygon, drag the vertices, use Add vertex or Remove vertex and follow the shoelace table. On Line forms, drag A and the tip of u, move Parameter t or press Run, and tick Second line d₂ to compare two lines; try the Tasks.
Parameters you can change
- Coefficient a (of x) -6–6
- Coefficient b (of y) -6–6
- Coefficient c -20–20
- Center I: x-coordinate -6–6
- Center I: y-coordinate -6–6
- Radius R 0.5–6
- Screen Line and circle, Two points, Polygon, Forms of a line equation
- Point A: x-coordinate -7–7
- Point A: y-coordinate -7–7
- Point B: x-coordinate -7–7
- Point B: y-coordinate -7–7
- Ratio AP : PB, part p 1–9
- Ratio AP : PB, part q 1–9
- d₁: point A, x-coordinate x₀ -7–7
- d₁: point A, y-coordinate y₀ -7–7
- d₁: direction vector, u₁ -6–6
- d₁: direction vector, u₂ -6–6
- Parameter t -4–4
- Show a second line d₂
- d₂: point B, x-coordinate -7–7
- d₂: point B, y-coordinate -7–7
- d₂: direction vector, v₁ -6–6
- d₂: direction vector, v₂ -6–6
Questions to explore
- If a, b and c are all multiplied by 2, do the line and the distance d(I, Δ) change? Why?
- For the circle shown, which values of Coefficient c make Δ tangent to (C)?
- Why are the vectors n = (a; b) and u = (−b; a) always perpendicular?
