# 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

1. Symmetry: For every value of x there are two equal and opposite values of y.
2. Region: For negative value of x then y value is imaginary.
∴ No part of curve lies to left of y-axis.
3. Origin: The curve passes through origin and tangent at origin is x=0 that is y – axis.
4. Intersection with axis: The curve meets.
5. Co – ordinate axis: The curve meets the co – ordinates axis only at origin.
6. 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,

$$SP=\sqrt{{{\left( x-a \right)}^{2}}+{{y}^{2}}}$$ $$=\sqrt{{{\left( x-a \right)}^{2}}+4ax}\left[ \because P\left( x,y \right)lies\,on\,parabola \right]$$ $$=\sqrt{{{\left( x+a \right)}^{2}}}$$

⇒ SP = a + x

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