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