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.
$P{S}^{2}=P{M}^{2}\Rightarrow {\left(xa\right)}^{2}+{y}^{2}={\left(x+a\right)}^{2}\Rightarrow {y}^{2}=4ax$
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 xaxis 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, yaxis 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, ${y}^{2}=4{a}^{2}\Rightarrow y=\pm 2a$. 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 Nonstandard Form
The horizontal and vertical parabolas with vertex other than the origin are:
${\left(yk\right)}^{2}=4a\left(xh\right)$ 
${\left(xh\right)}^{2}=4b\left(yk\right)$ 
Equation of the Parabola in Nonstandard 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 
L_{1}_{ }(a + h, 2a + k) and
L_{2}_{ }(a + h, −2a + k)

L_{1}( −2b + h, b + k) and
L_{2}_{ }(2b + h, b + k)

Parametric Equation of a Parabola
Condition for the General Equation of Second Degree to Represent a Parabola
The general equation of second degree $a{x}^{2}+2hxy+b{y}^{2}+2\mathrm{g\dots}$
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