The equations of the tangent and normal to the hyperbola at the point are and respectively.
Consider that the standard equation of a hyperbola with vertex at origin can be written as
Since the point lies on the given hyperbola, it must satisfy equation (i). So we have
Now differentiating equation (i) on both sides with respect to , we have
If represents 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 .