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).