# Standard form of Parabola and its Various Forms II

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

Some other standard forms of parabola:

A. y² = – 4ax 1. Co – ordinates of vertex = (a, 0)
2. Co – ordinates of focus = (-a, 0)
3. Equation of directrix = x – a = 0
4. Equation of axis is y = 0
5. Length of latus rectum = 4a
6. Focal distance = a – x

B. x² = 4ay 1. Co – ordinates of vertex = (0, 0)
2. Co – ordinates of focus = (0, a)
3. Equation of directrix is y + a = 0
4. Equation of axis is x = 0
5. Length of latus rectum = 4a
6. Length of focal distance to point P(x, y) = (a + y)

C. x² = – 4ay 1. Co – ordinates of vertex = (0, 0)
2. Co – ordinate of focus = (0, -a)
3. Equation of directrix is y – a = 0
4. Equation of axis is x = 0
5. Length of latus rectum = 4a
6. 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

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

$${{\left( \frac{ax+by+x}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)}^{2}}=\left( constant \right)\left( \frac{bx-ay+c’}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)$$

Put $$\frac{ax+by+x}{\sqrt{{{a}^{2}}+{{b}^{2}}}}$$ = Y and $$\frac{bx-ay+c’}{\sqrt{{{a}^{2}}+{{b}^{2}}}}$$= X

We get

y² = (constant) X

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