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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.
PS2=PM2 x-a2+y2=x+a2 y2=4ax
Thus, the standard equation of the horizontal parabola is y2=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 y2=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 L1L2, shown in the figure, is the latus rectum of the parabola y2=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 y2=4ax. Thus, y2=4a2 y=±2a. Therefore, the end points of the latus rectum are L1a, 2a and L2a,-2a.

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:

 
y-k2=4ax-h    
x-h2=4by-k 
                                                       
Equation of the Parabola in Non-standard Form 
              
The details regarding focus, vertex, directrix, etc. are:
       (yk)2 = 4a (xh) (xh)2 = 4b (yk)
Vertex (h, k) (h, k)
Focus (a + h, k) ( h, b + k)
Equation of the Directrix (xh) + a = 0 (yk) + b = 0
Equation of the axis y = k x = h
Tangent at the vertex x = h y = k
Equation of latus rectum xa = h yk = 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 y2=4ax is x=at2, y=2at. Thus, any point on the parabola y2=4ax can be taken as at2, 2at 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 ax2+2hxy+by2+2g…

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