Chord of Contact

The chord of contact refers to a specific line segment that is tangent to a curve at two distinct points. In the context of conic sections, such as ellipses, hyperbolas, and parabolas, the chord of contact is the line segment that is tangent to the conic at the points where lines drawn from a given external point touch the conic. This concept is particularly useful in coordinate geometry and can be applied to various problems involving conics.

Definition

Given a conic section (ellipse, hyperbola, or parabola) and a point ( P(x_1, y_1) ) outside the conic, the chord of contact is the line segment that connects the points of tangency of the tangents drawn from ( P ) to the conic.

Formulas

The equation of the chord of contact can be derived from the equation of the tangent to the conic at a general point ( (x, y) ) on the conic. For a given conic with equation ( f(x, y) = 0 ), the equation of the tangent at a point ( (x, y) ) is given by:

[ \left. \frac{\partial f}{\partial x} \right|{(x, y)} (X - x) + \left. \frac{\partial f}{\partial y} \right|{(x, y)} (Y - y) = 0 ]

where ( (X, Y) ) are the coordinates of any point on the tangent line.

For the chord of contact from a point ( P(x_1, y_1) ), we replace ( (x, y) ) with ( (x_1, y_1) ) in the equation of the tangent to get:

[ T: \left. \frac{\partial f}{\partial x} \right|{(x_1, y_1)} (X - x_1) + \left. \frac{\partial f}{\partial y} \right|{(x_1, y_1)} (Y - y_1) = 0 ]

Examples

Example 1: Ellipse

Consider an ellipse with the equation ( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ). The equation of the chord of contact from a point ( P(x_1, y_1) ) is given by:

[ T: \frac{x_1 X}{a^2} + \frac{y_1 Y}{b^2} = 1 ]

Example 2: Hyperbola

For a hyperbola with the equation ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ), the equation of the chord of contact from a point ( P(x_1, y_1) ) is:

[ T: \frac{x_1 X}{a^2} - \frac{y_1 Y}{b^2} = 1 ]

Example 3: Parabola

For a parabola with the equation ( y^2 = 4ax ), the equation of the chord of contact from a point ( P(x_1, y_1) ) is:

[ T: y_1 Y = 2a(X + x_1) ]

Differences and Important Points

Feature	Ellipse	Hyperbola	Parabola
Equation of Conic	( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 )	( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 )	( y^2 = 4ax )
Chord of Contact	( \frac{x_1 X}{a^2} + \frac{y_1 Y}{b^2} = 1 )	( \frac{x_1 X}{a^2} - \frac{y_1 Y}{b^2} = 1 )	( y_1 Y = 2a(X + x_1) )
Tangents	Two distinct real tangents	Two distinct real tangents	One real tangent
Points of Tangency	Lie on the ellipse	Lie on the hyperbola	Lie on the parabola
Application	Used in problems involving the ellipse and an external point	Used in problems involving the hyperbola and an external point	Used in problems involving the parabola and an external point

Conclusion

The chord of contact is a fundamental concept in the study of conic sections. It is used to find the equation of the line segment that is tangent to a conic at two points, where the tangents are drawn from an external point. Understanding this concept is crucial for solving problems in coordinate geometry, especially those related to tangents and normals to conic sections.