Two sides of a rhombus OABC ( lying entirely in first quadrant or fourth quadrant) of area equal to 2 sq. units, are `y =x/sqrt(3), y=sqrt(3)x` Then possible coordinates of B is / are ('O' being the origin)

Options

A. `(1+sqrt(3), 1+sqrt(3))`
B. `(-1-sqrt(3), -1-sqrt(3))`
C. `(3+sqrt(3), 3+sqrt(3))`
D. `(sqrt(3)-1, sqrt(3)-1)`

Answer

A, B

Explanation

`"Here," angleCOA = 30^(@).`
Area of rhombus `=2 xx (1)/(2) xx OA xx OC "sin" 30^(@)`
`"or " 2=(1)/(2)x^(2)`
or OA = OC =2
`"Also, "angleOAB = 150^(@)`
`therefore " cos"150^(@) = (OA^(2) +AB^(2)-OB^(2))/(2 OA xx AB)`
`OB^(2) = 8+4sqrt(3) " or " OB = sqrt(2)(sqrt(3)+1)`
Hence, the coordinates of B are `(+-sqrt(2)(sqrt(3)+1) " cos "45^(@), +-sqrt(2)(sqrt(3)+1)"sin" 45^(@)).`