a) The point (x,y) is equidistant from the circle x^2 +y^2 = 1 and the point (2,0). Show that (x,y) must lie on the curve (x-1)^2 +y^2 = 4(x-(5/4))^2. Show that this curve is a hyperbola.
b) If two tangents with slopes m1, m2 intersect at a point (X,Y) show that m1 and m2 must be the roots of the quadratic equation:
(a^2 - X) m^2 +2XY + (b^2 - Y^2) = 0
and deduce that if the tangents are perpendicular to each other, the point (X,Y) lies on a circle, centre the origin.
a) I am unsure how to show that the point MUST lie on the curve. All I can think of doing is finding a point that I know is equidistant between the curve and point, and then substituting these values into the equation of the curve? i.e. (1.5,0). Is this the correct way to approach this part of the problem.
To show that the curve is a hyperbola, do I just need to rearrange the equation they have given me into the form of a hyperbola ie. (x^2/a^2 - y^2/b^2 = 1) If so I have done this, and I have an equation that resembles a hyperbola, but I'm not sure this really SHOWS that the curve is a hyperbola. I think maybe I am missing something, any help would be great!
b) I don't know im totally stuck!