Let n be the number of points having rational coordinates at a constant distance from the point (0,3^1/2), then

a.n>2

b.n=1

c.n?=2

d.n=1

Let A(x₁, y₁) and B (x₂, y₂) be two points at equal distance d from (0, 3^1/2)

then 

(x₁-0)² + (y₁-0)² = d²..............(1)

(x²-0)² + (y²-0)² = d²..........(2)

from equation (1) and (2)

x₁² - x₂² +(y₁ - 3^1/2) - (y₂ - 3^1/2) =0

x₁² - x₂² + (y₁² - y₂²) -2*3^1/2 (y₁ -  y₂) =0

comparing rational and irrational parts on both sides

y₁ - y₂ =0   => y₁ = y₂ = a (suppose)

x₁² -x₂² + (y₁² - y₂²) =0 

x₁² = x₂² =b²

x₁ = x₂ = 

so points are 

so