Ron wrote:
. . .

An isotropic antenna placed at the focal point would collect all of
the rays whereas a directional antenna at would not.


Therefore, in this particular situation, the isotropic would have
higher gain and capture more power than any directional antenna.

Please correct me if I am wrong.


Not sure what you mean by "focal point", but the best any antenna can do
is to intercept half the energy in some equivalent cross-sectional area
of an impinging field. It does this when connected to a matched load.

When an antenna intercepts one watt from a field having a power density
of one watt per square meter, it's said to have an "effective aperture"
or "capture area" of one square meter. The higher the gain of an antenna
in some particular direction, the larger its effective aperture in that
direction. Consequently, a high gain antenna would "capture" more power
from a wave arriving in its favored direction than an isotropic antenna
would. It would, of course, capture less from other directions, but
assuming equal efficiency, both antennas would capture equal amounts
overall.

The "capture area" isn't some physical region with boundaries -- it's
simply a way of expressing how much power is extracted from a field
having a given power density. In other words, it's just another way of
expressing antenna gain.

Roy Lewallen, W7EL