Conics

Equation of a Parabola

Standard Equation of the Parabola

A parabola is defined as the locus of a point P equidistant from a fixed point (called focus) and a fixed line (called directrix). 

Let us consider origin (0, 0) as the vertex A of the parabola and let P (x, y) be any point on the locus.
Let the fixed point (focus) be S (a, 0) and the fixed line be x + a = 0.
Using the definition of parabola, we have PS = PM.
$P{S}^2=P{M}^2 \Rightarrow {\left(x-a\right)}^2+y^2={\left(x+a\right)}^2 \Rightarrow y^2=4ax$
Thus, the standard equation of the horizontal parabola is $y^2=4ax$.

Important Points and Lines Related to Parabola

• The constant ratio is called the eccentricity and is denoted by e. When the eccentricity is unity; e = 1, the conic is called a Parabola.
• ​ The line which passes through the focus and perpendicular to the directrix is called axis of the parabola. Here x-axis is the axis of the parabola $y^2=4ax$.
• The vertex of a parabola is defined as the intersection point of the parabola and its axis. Here (0, 0) is the vertex.
• The chord passing through the focus and perpendicular to the axis is called latus rectum. The chord $L_1{L}_2$, shown in the figure, is the latus rectum of the parabola $y^2=4ax$.
• Any chord which is perpendicular to the axis of the parabola is called double ordinate.
• The straight line passing through the vertex and perpendicular to the axis of the parabola is called tangent at vertex. Here, y-axis is the tangent at the vertex (0, 0).
• To find the end points of the latus rectum, put x = a in $y^2=4ax$. Thus, $y^2=4a^2 \Rightarrow y=\pm 2a$. Therefore, the end points of the latus rectum are $L_1\left(a, 2a\right) \mathrm{and} L_2\left(a,-2a\right)$${}_{\mathrm{}}$.

Other Horizontal and Vertical Parabolas

The following parabolas are drawn with their vertex, focus, etc.
                   
Equation of the Parabola in Non-standard Form
 
The horizontal and vertical parabolas with vertex other than the origin are:
 

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

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

                                                       
Equation of the Parabola in Non-standard Form 
              
The details regarding focus, vertex, directrix, etc. are:

	(y − k)2 = 4a (x − h)	(x − h)2 = 4b (y − k)
Vertex	(h, k)	(h, k)
Focus	(a + h, k)	( h, b + k)
Equation of the Directrix	(x − h) + a = 0	(y − k) + b = 0
Equation of the axis	y = k	x = h
Tangent at the vertex	x = h	y = k
Equation of latus rectum	x − a = h	y − k = b
Length of latus rectum	|4a|	|4b|
End points of latus rectum	L1 (a + h, 2a + k) and L2 (a + h,  −2a + k)	L1( −2b + h, b + k) and L2 (2b + h,  b + k)

Parametric Equation of a Parabola The parametric equation of the standard parabola $y^2=4ax$ is $x=a{t}^2, y=2at$. Thus, any point on the parabola $y^2=4ax$ can be taken as $\left(a{t}^2, 2at\right)$ and we call this point as point t.

Condition for the General Equation of Second Degree to Represent a Parabola

The general equation of second degree $a{x}^2+2hxy+b{y}^2+2\mathrm{g\dots }$