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, hyperbola, or parabola) at two points and is drawn from a given external point. This line is called the "chord of contact" because it contacts the conic at two points, which are the points of tangency.

Understanding the Chord of Contact

When a point lies outside a conic section, it is possible to draw two tangents to the conic from that point. The chord of contact is the line segment that connects the points of tangency on the conic.

For a Hyperbola

Consider a hyperbola with the equation:

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

Let ( P(x_1, y_1) ) be a point outside the hyperbola. The tangents from ( P ) to the hyperbola will touch the hyperbola at two distinct points. The line joining these points of tangency is the chord of contact.

Equation of the Chord of Contact

The equation of the chord of contact can be derived using the concept of tangents from an external point. For a hyperbola, the equation of the chord of contact drawn from a point ( P(x_1, y_1) ) is given by:

$$ \frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1 $$

This equation is obtained by replacing ( x^2 ) with ( x x_1 ) and ( y^2 ) with ( y y_1 ) in the standard equation of the hyperbola.

Differences and Important Points

Aspect	Chord of Contact for Hyperbola	Chord of Contact for Ellipse	Chord of Contact for Parabola
Standard Equation	( \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 )
Equation from ( P )	( \frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1 )	( \frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1 )	( y y_1 = 2a(x + x_1) )
Point of Tangency	Two distinct points	Two distinct points	One point
Nature of Conic	Open curve with two branches	Closed curve	Open curve with one branch

Examples

Example 1: Hyperbola

Given the hyperbola ( \frac{x^2}{16} - \frac{y^2}{9} = 1 ) and a point ( P(10, 6) ), find the equation of the chord of contact.

Solution:

The equation of the chord of contact from ( P(10, 6) ) is:

$$ \frac{x \cdot 10}{16} - \frac{y \cdot 6}{9} = 1 $$

Simplifying, we get:

$$ \frac{5x}{8} - \frac{2y}{3} = 1 $$

Multiplying through by 24 to clear the denominators:

$$ 15x - 16y = 24 $$

This is the equation of the chord of contact.

Example 2: Ellipse

Given the ellipse ( \frac{x^2}{25} + \frac{y^2}{16} = 1 ) and a point ( P(8, 6) ), find the equation of the chord of contact.

Solution:

The equation of the chord of contact from ( P(8, 6) ) is:

$$ \frac{x \cdot 8}{25} + \frac{y \cdot 6}{16} = 1 $$

Simplifying, we get:

$$ \frac{2x}{25} + \frac{3y}{8} = 1 $$

Multiplying through by 200 to clear the denominators:

$$ 16x + 75y = 200 $$

This is the equation of the chord of contact.

Example 3: Parabola

Given the parabola ( y^2 = 8x ) and a point ( P(2, 4) ), find the equation of the chord of contact.

Solution:

The equation of the chord of contact from ( P(2, 4) ) is:

$$ y \cdot 4 = 2 \cdot 8 (x + 2) $$

Simplifying, we get:

$$ y = 4x + 8 $$

This is the equation of the chord of contact.

In conclusion, the equation of the chord of contact is a powerful tool in conic section geometry, allowing us to find the line that touches the conic at two points from a given external point. The equation varies depending on the type of conic section, but the concept remains consistent across hyperbolas, ellipses, and parabolas.