# Find the Equation of the Tangent Line to the Hyperbola

Show that the always lies on the hyperbola . Find the equation of the tangent and normal to the hyperbola at the point .

We have standard equation of a hyperbola:

Putting and in equation (i), we have

Which is true for all values of , so the point always lies on the hyperbola (i).

Now differentiating equation (i) on both sides with respect to , we have

Let be the slope of the tangent at the given point , then

The equation of the tangent at the given point is

This is the equation of the tangent to the given hyperbola at .

The slope of the normal at is

The equation of the normal at the point is

This is the equation of the normal to the given hyperbola at .