Equation of the directrix and coordinates of the focus

Understanding Directrix and Focus of a Parabola

A parabola is a type of curve in geometry, which is defined as the locus of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
The directrix is a straight line that is not part of the parabola but is used to define its shape. The distance of each point of the parabola from the directrix is equal to its distance from the focus.
The focus is a particular point inside the parabola. The property of the focus is that any line drawn through it reflects off the parabola parallel to the directrix.

Equation of the Directrix

In the case where the vertex of the parabola is at the origin (0,0) and it opens either upwards or downwards, the equation of the directrix is y = -p.
If the vertex of the parabola is at the origin, but it opens either to the right or to the left, the equation of the directrix is x = -p.
If the vertex of the parabola is not at the origin, the equation will shift parallelly. For example, if the vertex of the parabola is (v1, v2) and it opens either upwards or downwards, the equation of the directrix is y = v2 - p.

Coordinates of the Focus

If the vertex of the parabola is at the origin (0,0) and the parabola is facing upwards, the coordinates of the focus are (0, p).
If the vertex of the parabola is at the origin and the parabola is facing downwards, the coordinates of the focus are (0, -p).
If the vertex of the parabola is at the origin and it opens towards the right or left, the focus is respectively at (p, 0) or (-p, 0).
If the vertex of the parabola is not at the origin, the position of the focus would adjust accordingly. E.g., if the vertex if the parabola is (v1, v2) and it opens either upwards or downwards, the focus is at (v1, v2+p) or (v1, v2-p).

Practising with the Formulae of Directrix and Focus

Consistent practice is key to understanding and applying the concepts of directrix and focus in parabola problems.
Try to solve several problems with parabolas placed at different positions to familiarise yourself with how the directrix and focus equations change based on the direction the parabola opens.
Always verify your answers using the definitions of the directrix and focus, ensuring the point on the parabola is equidistant from the focus and the directrix.

Through regular practice and familiarisation with these definitions and formulae, you’ll improve in understanding and working with different parabolic scenarios involving the directrix and focus.