The Coordinate Geometry of Curves

Essential Concepts

A curve in coordinate geometry usually refers to a locus of points satisfying a particular equation.
The equation of a circle centred at the origin in the (x, y) plane is given by x² + y² = r², where r is the radius of the circle.
For a circle centred at (h, k) in the plane, the equation is (x - h)² + (y - k)² = r².
An ellipse has an equation of the form (x²/a²) + (y²/b²) = 1, while a hyperbola is similar but has a minus sign, e.g. (x²/a²) - (y²/b²) = 1.
The term parabola refers to the graph of a quadratic function. It has an equation of the form y = ax² + bx + c or in the form y = a(x - h)² + k.

The centre of a circle, ellipse, or hyperbola is the midpoint of its major axis. For a parabola, it is the vertex.
The major axis of an ellipse or hyperbola is the line segment that cuts these curves at their widest points. The minor axis is the line segment at right angles to the major axis which bisects the curve.
The vertices of an ellipse or hyperbola are the points where the major axis intersects the curve.
In a parabola, the point called the focus is such that any point on the parabola is equidistant from the focus and the directrix.

A tangent to a curve at a particular point is the straight line that “just touches” the curve at that point.
A normal to a curve at a particular point is the line perpendicular to the tangent at that point.
The gradients of perpendicular lines multiply together to give -1, so the normal has gradient -1/m, if m is the gradient of the tangent.

The area enclosed by an ellipse with semi-major axis a and semi-minor axis b is given by πab.
The distance from a point (x₁, y₁) to the centre of a circle (h, k) is given by √((x₁ - h)² + (y₁ - k)²).
The length of an arc of a circle can be calculated using the formula rθ, where r is the radius and θ is the angle subtended at the centre, measured in radians.

Scaling a curve involves multiplying the x or y coordinates by a certain factor, stretching or compressing the curve.
Shifting a curve involves adding a constant to the x or y coordinates, moving the curve left/right or up/down.
To sketch a curve accurately, find key points such as intersections with the axes, maximum and minimum points, and points of inflection.
An inflection point is a point on a curve where the curve changes from being concave up (shaped like a U) to concave down (shaped like an n), or vice versa.