Let S be the set of all points `((a)/(b),(c)/(d))` on the circle with radius 1 centred at (0,0) where a and b are relatively prime integers, c and d are relatively prime integers (that is HCF (a, b) = HCF (c,d) = 1), and the integers b and d are even. Then the set S

Options

A. is empty
B. has four elements
C. has eight elements
D. is infinite

Answer

A

Explanation

`"circle is "x^(2)+y^(2)=1`
`y=pmsqrt(1-(a^(2))/(b^(2)))" "(becausex=(a)/(b))`
`y=pm(1)/(b)sqrt((b^(2))-a^(2))`
As y is retional so

`underset("even")underset(darr)(b^(2))-underset("odd")underset(darr)(a^(2))=underset("odd")underset(darr)(p^(2))`
`b^(2)=a^(2)+p^(2)`
`=(2k+1)^(2)+(2lamda+1)^(2)`
`=4k^(2)+4k+1+4lamda^(2)+4lamda+1`
`b^(2)=4(k^(2)+lamda^(2)+k+lamda)+2 " impossilbe"`
`"as L.H.S. is multiple of 4 but R.H.S is not multiple of 4 "`