PARABOLA -3-

Let us understand some other features of a parabola.

(a) Focal Distance:

The distance of a point on the parabola from its focus is called the focal distance of the point Focal distance of P = SP = x + a.

(b) Focal Chord:

A chord of the parabola, which passes through its focus, is called Focal chord.

(c) Latus Rectum:

The chord through focus and perpendicular to the axis of the parabola is called the latus rectum.

The co-ordinates of the end point of the latus rectum are (a, 2a) and (a, –2a) and length of latus rectum = 4a.


For horizontal parabola

Let us consider origin (0, 0) as the vertex A of the parabola and two equidistant points S(a, 0) as focus and Z(–a, 0) a point on the directrix now let P(x, y) be the moving point. Draw SZ perpendicular from S on the directrix. Then SZ is the axis of the parabola. Now the middle point of SZ, that is A, will lie on the locus of P.

i.e. AS = AZ

The x-axis along AS, and the y-axis along the perpendicular to AS, as A, as in the figure. Now by definition PM = PS ⇒ MP2 = PS2

So, that, (a + x)2 = (x – a)2 + y2.

Hence, the equation of horizontal parabola is y2 = 4ax.


Similarly for the vertical parabola

Let us consider origin (0, 0) as the vertex A of the parabola and two equidistant points S(0, b) as focus and Z(0, –b) a point on the directrix now let P(x, y) be the moving point. Draw SZ perpendicular from S on the directrix. Then SZ is the axis of the parabola. Now the middle point of SZ, that is A, will lie on the locus of P i.e. AS = AZ.

The y-axis along AS, and the x-axis along the perpendicular to AS at A, as in the figure.




Now by definition PM = PS

⇒ MP2 = PS2

So that, (b + y)2 = (y – b)2 + x2.

Hence, the equation of vertical parabola is x2 = 4by.


Finding the end points of latus Rectum

For finding the end points of latus rectum LL’ of the parabola y2 = 4ax, we put x = a as latus rectum passes through focus (a, 0) therefore we have

y2 = 4a2

⇒ y = + 2a

Hence the end points are (a, 2a) and (a, – 2a).

Also LSL’ = 2a – (–2a) = 4a = length of double ordinate through the focus S.


Note:

Two parabolas are said to be equal when their latus recta are equal.

The important points & lines related to standard Parabola

Note:

1. The points and lines of two parabolas can be interchanged by transformations.

2. If a > 0 & a <> 0 & b <>