Virginia state can be represented approximately on a

map as an area bounded by $39\mathrm{.}5$N$,36\mathrm{.}5$N$,76\mathrm{.}0$W

and $83\mathrm{.}0$W$.$ A geostationary satellite at $79\mathrm{.}5$W has

an antenna with a spot beam that covers all of Virginia

at a downlink centre frequency of $11\mathrm{.}155$ GHz$.$ In this

problem, assume an aperture efficiency of $65\%.$

a$)$ Determine the diameter of the antenna needed on the

satellite if it is a circular parabolic reflector that

generates a circular beam with a $3$dB beamwidth

equal to the diagonal of the area bounding the state of

Virginia. What is the gain of this antenna?

b$)$ If the satellite antenna is an elliptical parabolic reflector

with $3$dB beamwidths in the N$-$S and E$-$W directions that

are equal to the length and width of the area bounding

the state of Virginia, determine the dimensions of the

antenna in these directions and hence calculate its gain.