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Sunday, April 16, 2017

Standard form of Parabola and its various forms

Equation of parabola in its standard form: Let S be focus and ZZ’ be directrix of a parabola. Draw SK perpendicular from S on directrix and bisect SK at A

AS = SK

Where A lies on parabola

⇒ SK = 2a

SA = SK = a

Let choose A be origin, AS as X - axis and AY line perpendicular AS as Y - axis.

Then co - ordinates S = (a, 0) and equation of directrix is x + a = 0
 
P lies on parabola

⇒ SP = PM

⇒ SP² = PM²

(x - a)² + (y - 0)² = (x + a)²

y² + x² - 2ax + a² = x² + a² + 2ax

y² = 4ax

y² = 4ax is the standard equation of parabola

Tracing of parabola: We have y² = 4ax, a > 0

The equation can be written as y = k ± 2 √ (ax) we observe
  •  Symmetry: For every value of x there are two equal and opposite values of y.
  • Region: For negative value of x then y value is imaginary. No part of curve lies to left of y-axis.
  • Origin: The curve passes through origin and tangent at origin is x=0 that is y - axis.
  • Intersection with axis: The curve meets.
  • Co - ordinate axis: The curve meets the co - ordinates axis only at origin.
  • Portions occupied: As x → ∞, y → ∞. Curve extends to infinity, to the right of y - axis.
Various results related to parabola:

Double ordinate: let P be any point on parabola y² = 4ax. A chord passing through P perpendicular to axis of the parabola is called the double ordinate through point P.
Latus rectum: A double ordinate through focus is called the latus rectum.
Length of latus rectum = 4a.

Co - ordinates of < and <’ are (a, 2a) and (a, -2a) respectively.

Focal distant at any point: The distant P(x, y) from the focus S is called the focal distance of the point P.

Now,


SP = a + x.

a + x is the focal distance of point P(x, y).

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