Common Tangents to Hyperbolas: A Detailed Analysis

Introduction

This article provides a comprehensive guide to finding the equations of the common tangents to two hyperbolas given by the equations [frac{x^2}{a^2} - frac{y^2}{b^2} 1] and [frac{y^2}{a^2} - frac{x^2}{b^2} 1]. These hyperbolas are symmetric with respect to the line (y x) and the line (y -x). Understanding these tangents is fundamental to the study of algebraic geometry and calculus.

Equations of Tangents to Hyperbolas

The general form of the equation of a tangent to a hyperbola is derived using the point of tangency. For the hyperbola (frac{x^2}{a^2} - frac{y^2}{b^2} 1) at point ( (x_0, y_0) ), the equation of the tangent is given by:

[frac{x_}{a^2} - frac{y_0y}{b^2} 1]

Similarly, for the hyperbola (frac{y^2}{a^2} - frac{x^2}{b^2} 1) at point ( (x_1, y_1) ), the equation of the tangent is:

[frac{y_1y}{a^2} - frac{x_1x}{b^2} 1]

Identifying Common Tangents

To find the common tangents, we consider lines of the form (y mx c). Substituting this form into the equation of the first hyperbola:

[frac{x^2}{a^2} - frac{(mx c)^2}{b^2} 1]

This results in a quadratic equation in (x). For the line to be a tangent, the discriminant of this quadratic equation must be zero. Similarly, substituting the same form into the equation of the second hyperbola:

[frac{(mx c)^2}{a^2} - frac{x^2}{b^2} 1]

This also leads to a quadratic in (x) with the discriminant that must also be zero.

Final Result

The common tangents to both hyperbolas can be derived from the conditions on the discriminants. The equations of the common tangents are:

[y mx pm sqrt{a^2m^2 - b^2}]

where (m) is the slope of the tangent lines.

Symmetries and Tangents

Considering the symmetries of the hyperbolas, we can take the line (y x t) as a tangent to the first hyperbola. This gives:

[frac{x^2}{a^2} - frac{(x t)^2}{b^2} 1]

Expanding and simplifying, we get:

[frac{1}{a^2}x^2 - frac{1}{b^2}(x^2 2tx t^2) 1]

[left(frac{1}{a^2} - frac{1}{b^2}right)x^2 - frac{2t}{b^2}x - frac{t^2}{b^2} - frac{1}{a^2} 0]

The discriminant of this quadratic equation being zero gives:

[left(frac{2t}{b^2}right)^2 4left(frac{1}{a^2} - frac{1}{b^2}right)left(frac{t^2}{b^2} frac{1}{a^2}right) 0]

Solving for (t), we find:

[t pm sqrt{a^2 - b^2}]

Thus, the four common tangents to both hyperbolas are:

[y xs pm sqrt{a^2 - b^2}]

[y -x pm sqrt{a^2 - b^2}]

These tangents can be visualized using a sketch for specific values of (a) and (b), such as (a 3) and (b 2).

Conclusion

In conclusion, the equations of the common tangents to the hyperbolas (frac{x^2}{a^2} - frac{y^2}{b^2} 1) and (frac{y^2}{a^2} - frac{x^2}{b^2} 1) are:

(y mx sqrt{a^2m^2 - b^2}) (y mx - sqrt{a^2m^2 - b^2}) (y -x sqrt{a^2 - b^2}) (y -x - sqrt{a^2 - b^2}) (y x sqrt{a^2 - b^2}) (y x - sqrt{a^2 - b^2})

These equations provide a powerful tool for understanding the geometric properties and relationships between these hyperbolas.