## Coordinate Geometry

In the 17^{th} century Rene Descartes showed that algebraic methods could be applied to geometry and formulated the subject of coordinate geometry. The framework that we use when drawing pictures of curves is provided by a **Cartesian Coordinate System**. It consists of a pair of straight lines at right angles, called **coordinate axes** which may be supposed to be infinitely long. The point where they cross is the **origin**. The horizontal line is called the *x*-axis, and the vertical the *y*-axis. Distances to the right of the origin, along the *x*-axis are positive, those to the left, negative. Along the *y*-axis, distances above the origin are positive, below the origin, negative. Any pair of values (an **Ordered Pair**) for x and y can be used to determine a point in the coordinate system.

The **gradient-intercept form** of the equation of a straight line (a linear function) is,

*y* = *mx + c*

where *m *and *c* are constants, and *m* is the **gradient** (or **slope**) and c is the ** y-intercept**.

And it follows that *m* = (*y*_{2} – *y*_{1})/(*x*_{2} – *x*_{1})

The **point-gradient form **of a straight line is,

*y* – *y*_{1} = *m*(*x* – *x*_{1})

The** two-point form **of the equation of a straight line is,

(*y* – *y*_{1})/(* x* – *x*_{1}) = (*y _{2}* –

*y*

_{1})/(

*x*–

_{2}*x*

_{1})

The **intercept form** of the equation of a straight line is,

(*x/a*) + (*y/b*) = 1

Any line parallel to the *y*-axis has an equation *x = b* for some constant *b*.

The general equation of a line in two dimensions is,

*ax + by + c* = 0

Two lines of gradient *m*_{1} and *m*_{2} are perpendicular if *m*_{2}*m*_{1} = -1

The distance between two points *P* and *Q* is,

*PQ* = √[(*x*_{2} – *x*_{1})^{2} + (*y*_{2} – *y*_{1})^{2}]

The equation of a circle with Radius *R* is,

*x*^{2} + *y*^{2} = *R*^{2}

And the standard **equation of a circle** with centre (*x*_{0}, *y*_{0}) and radius *R* is,

(*x* – *x*_{0})^{2} + (*y* – *y*_{0})^{2} = *R*^{2}

The line *x*_{1}*x* + *y*_{1}*y* = *R*^{2}

Is a **tangent **to the circle *x*^{2} + *y*^{2} = *R*^{2} provided that the point (*x*_{1}, *y*_{1}) lies on the circle.

**Polar coordinates** are particularly useful when distances from a fixed origin are important. If the origin is represented by the point *O* and the point of interest is point *P*, the line segment from *O* to *P* (called the **radius vector** of *P*) is of length *r* and is inclined at an angle *θ* (measured in the anticlockwise direction), form an arbitrary chosen ray, called the **polar axis** that emanates from *O*. The length *r* is called the **radial coordinate**, and *θ* the **polar angle** of *P*.

The polar coordinates *r* and *θ* can be combined so that the point *P* can be represented by the ordered pair (*r*, *θ*).

Cartesian and Polar coordinates are linked using,

*x* = *r* cos *θ *and *y* = *r* sin *θ
*

It can also be seen that,

*r*^{2} = *x*^{2} + *y*^{2}, sin* θ* = *y/r* and cos *θ* = *x/r*

To remove and ambiguity about the choice of direction of a *z*-axis, a **right-handed Cartesian coordinate system** is always used. If the thumb and first two fingers of the right hand are arranged so that they are mutually perpendicular, the first and second finger point along the *x* and *y*-axis, the thumb along the *z*-axis.

The distance *d*, between two points (*x*_{1}, *y*_{1}, *z*_{1}) and (*x*_{2}, *y*_{2}, *z*_{2}) is given by,

*d*^{2} = (*x*_{2} – *x*_{1})^{2} + (*y*_{2} – *y*_{1})^{2} + (*z*_{2} – *z*_{1})^{2}

The general **equation of a plane** in three dimensions is often written as,

*ax* + *by* + *cz* = *d*

where *a, b, c* and *d* are constants.

In general the **equations of a line in three dimensions**,

*(x – a)/l = (y – c)/m = (z – d)/n
*

specify a line through the point (*a, b, c*) and the constants *l, m* and *n* are known as the **direction ratios** (or **direction cosines**) of the line.