Condition for Focal Chord

In the study of conic sections, particularly parabolas, a focal chord is a line segment that passes through the focus of the parabola and has both endpoints on the parabola itself. Understanding the condition for a chord to be a focal chord is important for solving various problems related to parabolas.

Definition of a Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard equation of a parabola with its vertex at the origin and axis of symmetry along the x-axis is:

$$ y^2 = 4ax $$

where ( a ) is the distance from the vertex to the focus (and also to the directrix).

Focal Chord

A focal chord of a parabola is a line segment that passes through the focus and whose endpoints lie on the parabola. For the standard parabola ( y^2 = 4ax ), the focus is at ( (a, 0) ).

Condition for Focal Chord

For a chord to be a focal chord of the parabola ( y^2 = 4ax ), the product of the x-coordinates of the endpoints of the chord must be equal to ( a^2 ). This is because the focal distance (distance from the focus to a point on the parabola) for any point ( (x, y) ) on the parabola is ( x + a ).

If ( P(t_1) ) and ( Q(t_2) ) are two points on the parabola ( y^2 = 4ax ) with parameters ( t_1 ) and ( t_2 ) respectively, then the coordinates of ( P ) and ( Q ) are given by:

$$ P(t_1) = (at_1^2, 2at_1) $$ $$ Q(t_2) = (at_2^2, 2at_2) $$

For ( PQ ) to be a focal chord, the following condition must be satisfied:

$$ at_1^2 \cdot at_2^2 = a^2 $$

Simplifying, we get:

$$ t_1 \cdot t_2 = 1 $$

This is the condition for ( PQ ) to be a focal chord of the parabola ( y^2 = 4ax ).

Table of Differences and Important Points

Property	Non-Focal Chord	Focal Chord
Definition	A chord that does not pass through the focus.	A chord that passes through the focus of the parabola.
Equation	No specific condition on the product of the x-coordinates.	Product of the x-coordinates of the endpoints is equal to ( a^2 ).
Parametric Condition	No specific condition on the parameters ( t_1 ) and ( t_2 ).	( t_1 \cdot t_2 = 1 ) for the endpoints of the chord.
Example	Any chord that is not a focal chord.	The latus rectum, which is a line segment perpendicular to the axis of symmetry and passes through the focus.

Examples

Example 1: Latus Rectum as a Focal Chord

The latus rectum is a special case of a focal chord. It is the line segment perpendicular to the axis of symmetry of the parabola and passes through the focus. For the parabola ( y^2 = 4ax ), the latus rectum has endpoints at ( (a, 2a) ) and ( (a, -2a) ). Clearly, the product of the x-coordinates is ( a \cdot a = a^2 ), satisfying the condition for a focal chord.

Example 2: Finding a Focal Chord

Given the parabola ( y^2 = 8x ), find the endpoints of a focal chord.

Let the parameter of one endpoint be ( t_1 ). Then, by the condition for a focal chord, the parameter of the other endpoint must be ( t_2 = \frac{1}{t_1} ).

If ( t_1 = 2 ), then ( t_2 = \frac{1}{2} ). The coordinates of the endpoints are:

$$ P(t_1) = (8t_1^2, 16t_1) = (32, 32) $$ $$ Q(t_2) = (8t_2^2, 16t_2) = (2, 8) $$

Thus, the line segment joining ( (32, 32) ) and ( (2, 8) ) is a focal chord of the parabola ( y^2 = 8x ).

Understanding the condition for a focal chord is crucial for solving problems related to parabolas in mathematics, especially in the context of coordinate geometry and calculus.