The tangent at on the hyperbola has equation .
If teh tangent at P is also tangent to the circle center and radius , show that where e is the eccentricity of the hyperbola.
You then solve the problem by saying that a line is a tangent to a circle if and only if the distance from the centre of the circle to the line is equal to the radius of the circle. So in this case we get (using the distance formula with the point and the line ):
, using and
So the distance from to the line is . We want this to be equal to . That gives the equation
Simplify the big square root on the right-hand side of that equation as much as you can, using the relations and . Then square both sides of the equation, and solve it for . You should find that there are two solutions, and . But the first of those solutions cannot occur, because and .