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

We have standard equation of 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) both sides with respect to , we have

Let be 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 .