Equation of Chord of Contact

The equation of the chord of contact refers to the equation of the line segment that touches a conic section (ellipse, parabola, or hyperbola) at two distinct points. This line is formed by connecting the points of tangency from a given external point to the conic section. In this content, we will focus on the ellipse, but the concept can be extended to other conic sections as well.

Understanding the Chord of Contact

For an ellipse, the chord of contact is the line segment that touches the ellipse at two points where the tangents from an external point meet the ellipse. This line is significant in the study of conic sections and has applications in geometry and physics.

Equation of the Chord of Contact for an Ellipse

Given an ellipse with the equation:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

where (a) is the semi-major axis and (b) is the semi-minor axis, and a point (P(x_1, y_1)) outside the ellipse, the equation of the chord of contact can be derived using the concept of tangents from an external point.

The equation of the tangent to the ellipse at any point ((x, y)) on it is given by:

$$ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 $$

Since the chord of contact is formed by the tangents from the external point (P), its equation is the same as the combined equation of the tangents passing through (P):

$$ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 $$

This is the equation of the chord of contact for the point (P(x_1, y_1)) with respect to the given ellipse.

Important Points and Differences

Aspect	Description
Definition	The chord of contact is the line segment that touches a conic section at two points of tangency.
Equation (Ellipse)	For an ellipse (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1), the equation is (\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1) where (P(x_1, y_1)) is an external point.
Significance	The chord of contact is useful in determining the locus of points from which tangents to the ellipse make a constant angle.
Application	Used in problems involving tangents, normals, and properties of an ellipse.

Examples

Example 1: Find the Equation of the Chord of Contact

Given an ellipse (\frac{x^2}{9} + \frac{y^2}{4} = 1) and a point (P(6, 8)), find the equation of the chord of contact.

Solution:

Using the formula for the chord of contact, we have:

$$ \frac{x \cdot 6}{9} + \frac{y \cdot 8}{4} = 1 $$

Simplifying, we get:

$$ \frac{2x}{3} + 2y = 1 $$

Multiplying through by 3 to clear the fraction:

$$ 2x + 6y = 3 $$

This is the equation of the chord of contact for the point (P(6, 8)) with respect to the given ellipse.

Example 2: Application in Problem Solving

Suppose we want to find the locus of points from which tangents to the ellipse (\frac{x^2}{25} + \frac{y^2}{16} = 1) make a right angle.

Solution:

The condition for the tangents to be perpendicular is that the product of their slopes is -1. Let the point be (P(x_1, y_1)). The slopes of the tangents from (P) to the ellipse are given by the derivative of the chord of contact equation:

$$ \frac{d}{dx} \left(\frac{xx_1}{25} + \frac{yy_1}{16}\right) = 0 $$

Differentiating implicitly and setting the product of the slopes to -1, we can solve for the locus of (P). This will yield a second-degree equation representing another conic section, which is the required locus.

By understanding the equation of the chord of contact and its applications, students can solve a variety of problems related to conic sections, particularly ellipses, in their exams.