## Parabola – Analytic geometry Formulas

### Parabola

Parabola is the locus of point that moves such that it is always equidistant from a fixed point and a fixed line. The fixed point is called focus and the fixed line is called directrix. ### Parabola with vertex at the origin:

Parabola with vertex at the origin and open to the right.

${y}^{2}=4ax$

Parabola with vertex at the origin and open to the left.

${y}^{2}=-4ax$

Parabola with vertex at the origin and open upward.

${x}^{2}=4ay$

Parabola with vertex at the origin and open downward.

${x}^{2}=-4ay$

### Parabola with vertex at any point (h, k)

Parabola with vertex at (h, k) and open to the right.

$\left(y-k{\right)}^{2}=4a\left(x-h\right)$

Parabola with vertex at (h, k) and open to the left.

$\left(y-k{\right)}^{2}=-4a\left(x-h\right)$

Parabola with vertex at (h, k) and open upward.

$\left(x-h{\right)}^{2}=4a\left(y-k\right)$

Parabola with vertex at (h, k) and open downward.

$\left(x-h{\right)}^{2}=-4a\left(y-k\right)$

### Example:

Find the equation of the parabola having vertex (0, 0), axis along the x-axis and passing through (2, −1).

${y}^{2}=4px$

We need to find p. We know the curve goes through (2,−1), so we substitute:

$\left(–1{\right)}^{2}=4\left(p\right)\left(2\right)$
$1=8p$
$p=\frac{1}{8}$
${y}^{2}=\frac{x}{2}$

### Elements of Parabola:

1. Focus:

Focus is located at distance a from vertex in the direction of parabola’s opening.

1. Directrix:

Directrix is at distance a from the vertex. It is a straight line located at the opposite side of parabola’s opening.

1. Vertex:

Vertex is the point extremity of parabola, i.e. highest point for open downward, lowest point for open upward, rightmost point for leftward, and leftmost point for rightward. The coordinates of vertex is denoted as (h, k).

1. Axis:

Axis is the line of symmetry of parabola. It contains both the focus and the vertex and always perpendicular to the directrix.

1. Latus Rectum:

Latus Rectum, denoted by LR, is a line perpendicular to the axis, passing through the focus and terminates on the parabola itself. The total length of LR is 4a (LR=4a), where aa stands for the distance from focus to vertex.

1. Eccentricity:

Eccentricity of parabola is always equal to 1 (e=1). Thus, parabola can also be defined as a conic section of eccentricity equal to 1.

### Example:

Find the focus for the equation y2=5x

Converting y2 = 5x to y2 = 4ax form, we get y2 = 4 (5/4) x

so a = 5/4, and the focus of y2=5x is:

F = (a,0) = (5/4,0)