Let n be the number of points having rational coordinates equidistant from the point `(0,sqrt3)`, then

Options

A. `ngt2`
B. `nle1`
C. `nle2`
D. `n=1`

Answer

C

Explanation

Let `A(x_1,y_1)` and `B(x_2,y_2)` be two points at equal distance a from `P(0,sqrt(3))`. Then
`x_1^(2)+(y_1-sqrt(3))^(2)=a^2` (1)
and `x_(2)^(2)+(y_2-sqrt(3))^(2)=a^2` (2)
Subtracting (2) from (1) we get
`x_1^(2)-x_(2)^2+y_1^(2)-y_(2)^(2)-2sqrt(3)(y_1-y_2)=0`
Equating irrational and rational parts to zero, we get
`y_1=y_2`
and `x_1^(2)-x_(2)^(2)+y_1^(2)-y_(2)^(2)=0`
thus, `y_1=y_2=a` (say)
and `x_1^(2)=x_2=b^2` (say)
Hence, at most two such points are possible whose coordinates are `(b,a) and (-b,a)`.
In a special case, when `b=0`, only one such point is possible.