My question(s):

For an example, on the 10 meter band:
If I take a 28" whip and mount it at the end of helical wound coil (wound on
1" diameter form), where the "wire length" of the coil, PLUS, the length of
the whip (28" + coil wire length) is equal to 1/2 wavelength (electrical
length)--BUT, the overall physical length of the antenna (top, tip of whip
to base of helical wound coil) is 1/4 wavelength, what would the radiation
pattern of such an antenna be?

Would it favor the pattern of a 1/4 or 1/2 wave antenna--or, would the
pattern be a compromise between the two--or, would the pattern be totally
unrelated to either?

What could I expect the impedance of such an antenna be? Would the
reactance be capacitive or inductive? What would be the best way to provide
a match to 50 ohm coax from such an antenna?

What software is available to model such an antenna?

Thanks in advance,
warmest regards