If poitns `A(3,5)` and B are equidistant from `H(sqrt2,sqrt5)` and B has rational coordinates,then `AB=` (A) `sqrt(7)` (B) `sqrt((3-sqrt2)^2+(5-sqrt5)^2)` (C) `ssqrt(34)` (D) none of these

Options

A. `sqrt(7)`
B. `sqrt((3-sqrt2)^2+(5-sqrt5)^2)`
C. `ssqrt(34)`
D. none of these

Answer

D

Explanation

Let `B-=(alpha,beta)`, where `alpha,beta` are rational.
Given `HA^2=HB^2`
`rArr(3-sqrt2)^2+(5-sqrt5)^2=(alpha-sqrt2)^2+(beta-sqrt5)^2`
`41-6sqrt2-10sqrt5=alpha^2+beta^2+7-2sqrt2alpha-2sqrt5beta`
`rArr alpha^2+beta^2-34++2(3-alpha)sqrt2+2(5-beta)sqrt5=0`
Since `alpha` and `beta` are rational,
`3-alpha=0rArralpha=3`
and `5-beta=0rArrbeta=5`
`therefore AB=0`