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 .