finding parametric equations

[aegis]

I have the following graph:
http://i45.tinypic.com/14c5o2u.png

and I am to find paramettric equations for
the curve that consists of all possible positions
of the point P in the image above, using the
angle theta as the parameter. Also note,
the line segment AB is tangent to the larger circle.
(The inner circle is supposed to be that, a circle --
apologies for making it look like an ellipse).

point A can be defined by the following ordered
pair: (a cos t, a sin t) where t stands for theta.

My book says point B is the ordered pair (a sec t, 0)
but I do not see how. The slope of the line segment
OB(where O represents origin) is: (sin t)/(cos t)
or (tan t) and the derivative of (tan t) is (sec^2 t)
and the slope of the line segment AB would then
have slope (sec^2 t) for some t, but this does not
lead me to point B being defined by the ordered pair
(a sec t, 0)

Where am I screwing this up?

[Lynne Vickson]

Look at the triangle OAB with right angle at point A. If x = |OB| we
have x*cos(t) = a, so x = a/cos(t).

R.G. Vickson