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Standard Equation of Parabola

Consider the focus of the parabola as S(a, 0) and directrix be x + a = 0, and axis as x-axis.
 
71572.png
 
Now according to the definition of the parabola for any point on the parabola, we must have SP = PM
70974.png = PN + NM = x + a
(xa)2 + y2 = (a + x)2
y2 = (a + x)2 – (xa)2
y2 = 4ax
 
Vertex:
(0, 0)
Tangent at vertex:
x = 0
Equation of latus rectum:
x = a
Extremities of latus rectum:
P(a, 2a), Q(a, – 2a)
Length of latus rectum:
4a
Focal distance (SP):
SP = PM = x + a
Parametric form:
x = at2 and y = 2at, where t is parameter

Other Standard Forms of Parabola

70953.png
 
Equation of curve y2 = –4ax
Vertex (0, 0)
Focus (–a, 0)
Directrix x – a = 0
Axis y = 0
Tangent at vertex x = 0
Parametric form (–at2, 2at)
 
70947.png
 
Equation of curve x2 = 4ay
Vertex (0, 0)
Focus (0, a)
Directrix y + a = 0
Axis x = 0
Tangent at vertex y = 0
Parametric form (2atat2)

70941.png
 
Equation of curve x2 = –4ay
Vertex (0, 0)
Focus (0, –a)
Directrix y – a = 0
Axis x = 0
Tangent at vertex y = 0
Parametric form (2at, –at2)

Equation of parabola when vertex is (h, k) and axis is parallel to the x-axis.
 
(yk)2 = 4a(xh)
 
70928.png
 
Equation of parabola when vertex is (h, k) and axis is parallel to the y-axis.
 
(xh)2 = 4a(yk)
 
70922.png




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