Standard form of Parabola and its Various Forms II

Focal chord: A chord of parabola is a focal chord, if it passes through focus.

Some other standard forms of parabola:

A.  y² = - 4ax

• Co - ordinates of vertex = (a, 0).
• Co - ordinates of focus = (-a, 0).
• Equation of directrix = x - a = 0.
• Equation of axis is y = 0.
• Length of latus rectum = 4avi.
• Focal distance = a - x.

B. x² = 4ay

• Co - ordinates of vertex = (0, 0).
• Co - ordinates of focus = (0, a).
• Equation of directrix is y + a = 0.
• Equation of axis is x = 0.
• Length of latus rectum = 4a.
• Length of focal distance to point P(x, y) = (a + y).

C. x² = - 4ay

• Co - ordinates of vertex = (0, 0).
• Co - ordinate of focus = (0, -a).
• Equation of directrix is y - a = 0.
• Equation of axis is x = 0.
• Length of latus rectum = 4a.
• Focal distance at a point P(x, y) = a - y.

Here is a table which are all together

		y² = 4ax	y² = - 4ax	x² = 4ay	x² = - 4ay
1	Co - ordinates of vertex	(0, 0)	(0, 0)	(0, 0)	(0, 0)
2	Co - ordinates of focus	(a, 0)	(-a, 0)	(0, a)	(0, -a)
3	Equation of directrix	X = - a	X = a	Y = -a	Y = a
4	Equation of axis	Y = 0	Y = 0	X = 0	X = 0
5	Length of latus rectum	4a	4a	4a	4a
6	Focal distance of a point P(x, y)	a + x	a - x	a + y	a - y

If vertex of a parabola at a point A (h, k) and its latus – rectum of length 4a then its equation is

• (y - K)² = 4a (x - h), if it axis is parallel to OX that is parabola open right ward.
• (y - K)² = - 4a (x - h), if its axis is parallel to OX’, that is opens leftward.
• (x - h)² = 4a (y - k), if its axis is parallel to OY that is opens upward.
• (x - h)² = - 4a (y - k), if its axis is parallel to OY’ that is opens downward.

Parametric equation of parabola: co - ordinates of any point on parabola y² = 4ax is (at², 2at) where t ϵ R.

The equation x = at², y = 2at taken together are called the parametric equation of parabola.

The parametric equation of (y - k)² = 4a (x - h) are x = h + at², y = k + 2at.

parabola	y² = 4ax	y² = - 4ax	x² = 4ay	x² = - 4ay
Parametric co -ordinates	(at², 2at)	(-at², 2at)	(2at, at²)	(2at, - at²)
Parametric equation	x = at², y = 2at	x = - at², y = 2at	x = 2at, y = at²	x = 2at, y = -at²

If suppose the equation of parabola is quadratic in both x and y, then to find its vertex focus, axis.

First obtain the equation of parabola and express it in here form (ax + by + c)² = [Constant] (bc - ay + c’). If should be noted here that ax + by + c & bx + ay + c are perpendicular lines.

Divide both sides by √ (a² + b²) to obtain

Put 

We get

y² = (constant) X

Compare y² = 4ax to obtain vertex, focus, axis etc.