# The Coordinate Geometry of Curves

## 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.

## Curve Characteristics

- 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.

## Tangents and Normals

- 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.

## Areas and Distances

- 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.

## Transformations and Curve Sketching

- 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.