Parabola








Parabola

Definitition 1
Parabola is conic section defined as a plane curve obtained by intersection of a cone and a plane in which
the plane misses the vertex and
the plane is parallel to the generator
Definition 2
Parabola is conic section defined as a plane curve obtained by intersection of a cone and a plane in which if
α= angle between generator and axis and
β= angle between plane and axis,
and
α=β
then
the section is a parabola,
in which
eccentricity = \(\frac{\cos \beta}{\cos \alpha}\)
here
the eccentricity is the measure of how far the conic deviates from being circular
Definition 3
Parabola is a plane curve defined a locus of a point in which
\( \frac{\text{distance from a fixed point}}{\text{distance from a fixed line}} \)= constant (e=1)

In this definition of conic section,
the constant ratio is called eccentricity, it is denoted by e.
the fixed point is called focus.
the fixed line is called directrix.

Thus
Parabola is a plane curve defined a locus of a point in which the distance from a fixed point (or focus) and distance from a fixed line (or directrix) is always equal.
Parabola is a plane curve defined a locus of a point which is always equidistant from a fixed point (or focus) to a fixed line (or directrix)
In this definition
  1. Focus: The fixed point of parabola is called focus
  2. Directrix: The fixed line of parabola is called directrix
  3. Axis: The straight line passing through focus and perpendicular to directrix is called axis
  4. Vertex: The meeting point of axis and parabola is called vertex
  5. Latus rectum: the chord passing through focus and perpendicular to axis is called latus rectum.
    The distance between the meeting points of latus rectum to the parabola is called length of latus rectum.



Standard Equation of Parabola

Let C be a parabola whose
Focus is F (a,0)
Directrix is \(l: x = -a\)
Vertex is O: (0,0)
Take any point P(x,y) on parabola C,

Then
  1. Draw PA ⊥ \(l\) then A (-a,y)
  2. Join F and P

By the definition of parabola
PA = PF
or \( (x+a)^2=(x-a)^2+y^2 \)
or \(x^2+2ax+a^2=x^2-2ax+a^2+y^2\)
or \(2ax=-2ax+y^2 \)
or \(y^2=4ax \)

Summary on Equation of Parabola

The basic parameters of parabola are summarized as below

Parabola Parabola Parabola Parabola Parabola
Equation \( y^2 =4 a x \) \( x^2 =4 a y \) \( (y-k)^2 =4 a (x-h) \) \( (x-h)^2 =4 a (y-k) \)
Vertex (0,0) (0,0) (h,k) (h,k)
Focus (a,0) (0,a) (h+a,k) (h,k+a)
Directrix x=-a y=-a x=h-a y=k-a
Axis y=0 x=0 y=k x=h
Axis of Symmetry x-axis y-axis y=k x=h
Endpoints of Latus Rectum (a,±2a) (±2a,a) (h+a,k±2a) (h±2a,k+a)



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