Relationship Between a Hyperbola and a Line

For the hyperbola and the line , the discriminant of the quadratic equation formed by eliminating determines the nature of their intersection:

1. The line intersects the hyperbola at two distinct points.
2. The line is tangent to the hyperbola, touching at exactly one point.
3. The line does not intersect the hyperbola.

Equation of the Tangent Line to a Hyperbola

1. When the slope is given
   For the hyperbola , the equation of a tangent line with slope is:
   where
   
   
   For the hyperbola , the equation of a tangent line with slope is:
   where
   
   
2. When the coordinates of the point of tangency are given
   For the hyperbola , the equation of the tangent line at the point is:
   
   For the hyperbola , the equation of the tangent line at the point is:

Equation of the Tangent Line from a Point Outside the Hyperbola

1. [Method 1] Assume the slope of the tangent line, and use the formula for the equation of the tangent line when the slope is given.
   
   
2. [Method 2] Assume the coordinates of the point of tangency , and use the formula for the equation of the tangent line when the point of tangency is given.