Page 28 - 12 Math 6

16.. CQounaicdSraetcitcioEnqsuations eeLLeeaarrnn..PPuunnjjaabb

Definitions

(i) The line through the focus and perpendicular to the directrix is called axis of the

parabola. In case of (2), the axis is y = 0.

(ii) The point where the axis meets the parabola is called vertex of the parabola. Clearly

the equation (2) has vertex A(0,0). The line through A and perpendicular to the axis

of the parabola has equation x = 0. It meets the parabola at coincident points and so

it is a tangent to the curve at A.

(iii) A line joining two distinct points on a parabola is called a chord of the parabola.

A chord passing through the focus of a parabola is called a focal chord of the

parabola. The focal chord perpendicular to the axis of the parabola (1) is called

latusrectum of the parabola. It has an equation x = a and it intersects the curve at

the points where

y2 = 4a2 or y = ±2a

Thus coordinates of the end points L and L’ of the latusrectum are

L(a, 2a) and L′(a, -2a).

The length of the latusrectum is LL′ = 4a.

(iv) The point (at2 , 2at) lies on the parabola y2 = 4ax for any real t.

x = at2 , y = 2at

are called parametric equations of the parabola y2 = 4ax.

6.4.1 General Form of an Equation of a Parabola.

Let F(h,k) be the focus and the line lx + my + n =0 be the directrix of a parabola. An
equation of the parabola can be derived by the definition of the parabola . Let P(x , y) be a
point on the parabola. Length of the perpendicular PM from P(x , y) to the directix is given by;

PM = lx + my + n
l2 + m2

By definition, (x - h)2 + (y - k)2 =(lx +l2 my + n)2
+ m2

is an equation of the required parabola.

A second degree equation of the form

ax2 + by2 + 2gx + 2fy + c = 0

version: 1.1

28