Standard equation of an ellipse with major axis on #x# axis (as origin is at center and one focus too is on #x# axis) is
#x^2/a^2+y^2/b^2=1# in which #b^2=a^2(1-e^2)#, and #e# is eccentricity. In such an equation focii are at #(ae,0)# and #(-ae,0)#.
As #ae=sqrt3#, #b^2=a^2-a^2e^2 = a^2-3#
Hence equation reduces to the form
#x^2/a^2+y^2/(a^2-3)=1# and putting #a^2=p#, this reduces to
#x^2/p+y^2/(p-3)=1# and as it passes through #(1, sqrt3/2)#, we get
#1/p+3/(4(p-3)}=1# which simplifies to #4p^2-19p+12=0# i.e. #(p-4)(4p-3)=0# and gives value of #p# or #a^2# as #4# or #3/4#. As #a>sqrt3# (focii being at #(-sqrt3,0)# and #(sqrt3,0)#), only possible value is #a=2#.
Hence #b^2=a^2-a^2e^2= 4-3=1#
and equation of ellipse is #x^2/4+y^2=1#
