Question

Let `x_1,x_2,x_3,x_4` be four non zero numbers satisfying the equation `tan^-1 (a/x)+tan^-1(b/x)+tan^-1(c/x)+tan^-1(d/x)=pi/2` then which ofthe following relation(s) hold good?
A. `x_(1)+x_(2)+x_(3)+x_(4)=a+b+c+d`
B. `(1)/(x_(1))+(1)/(x_(2))+(1)/(x_(3))+(1)/(x_(4))=0`
C. `x_(1)x_(2)x_(3)x_(4)=abcd`
D. `(x_(2)+x_(3)+x_(4))(x_(3)+x_(4)+x_(1))(x_(1)+x_(2)+x_(3))=abcd`

Answer 1

Correct Answer - B::C::D
`"tan"^(-1)(a)/(x)+"tan"^(-1)(b)/(x)+"tan"^(-1)(c )/(x)+"Tan"^(-1)(d)/(x)=(pi)/(2)`
`rArr "tan"^(-1)(a)/(x)"tan"^(-1)(b)/(x)=(pi)/(2)-("tan"^(-1)(c )/(x)+"tan"^(-1)(d)/(x))`
`rArr "tan"^(-1)((a)/(x)+(b)/(x))/(1-(ab)/(x^(2)))=(pi)/(2)-"tan"^(-1)((c )/(x)+(d)/(x))/(1-(cd)/(x^(2)))`
`rArr "tan"^(-1)((a+b)x)/(x^(2)-ab)="cot"^(-1)((c+d)x)/(x^(2)-cd)`
`rArr ((a+b)x)/(x^(2)-ab)=(x^(2)-cd)/((c-d)x)`
`rArr x^(4)-(ab+ac+ad+bc+bd+cd)x^(2)+abcd=0`
This equation has roots `x_(1),x_(2),x_(3),x_(4)`
`therefore Sigma x_(1)=0, Sigma x_(1)x_(2)=- Sigma ab, Sigma x_(1)x_(2)x_(3)=0` and `x_(1)x_(2)x_(3)x_(4)=abcd`
`therefore sum(1)/(x_(1))=0`
and `(x_(2)+x_(3)+x_(4))(x_(3)+x_(4)+x_(1))(x_(4)+x_(1)+x_(2))(x_(1)+x_(2)+x_(3))`
`=x_(1)x_(2)x_(3)x_(4)`