For the last part, let P be the point . The tangent satisfies , and it passes through P if . Substitute n from the second of those two equations into the first equation: , which can be written as

The two roots of the quadratic equation (*) in are and , where are the gradients of the tangents through P (because is the gradient of the line ). Therefore (formula for product of roots of quadratic: constant term divided by coefficient of ). But you already know that . Put those two expressions for together to see that , from which . Finally, replace by to get the locus of P.