If the circle `x^2+y^2=a^2` intersects the hyperbola `xy=c^2` in four points `P(x_1,y_1)`,`Q(x_2,y_2)`,`R(x_3,y_3)`,`S(x_4,y_4)`, then which of the following need not hold.
(a)  `x_1+x_2+x_3+x_4=0`
(b)  `x_1 x_2 x_3 x_4=y_1 y_2 y_3 y_4=c^4`
(c)  `y_1+y_2+y_3+y_4=0`
(d)  `x_1+y_2+x_3+y_4=0`

Options

A. `x_(1)+x_(2)+x_(3)+x_(4)=01`
B. `y_(1)+y_(2)+y_(3)+y_(4)=0`
C. `x_(1)x_(2)+x_(3)x_(4)=c^(4), y_(1)y_(2)y_(3)y_(4)=c^(4)`
D. all of these

Answer

D

Explanation

The x-coordinates of P, Q, R and S are the roots of the equation
`x^(2)+((c^(2))/(x))^(2)=a^(2)rArr x^(4)+0x^(3)-a^(2)x^(2)+0x+c^(4)=0`
`:. x_(1)+x_(2)+x_(3)+x_(4)=0 and x_(1)x_(2)x_(3)x_(4)=c^(4)`
Similarly , y-coordinates are the roots of the equation
`y^(2)+((c^(2))/(y))^(2)=a^(2) rArr y^(4) + 0y^(3)-a^(2)y^(2)+0y+c^(4)=0`
This give that `y_(1)+y_(2)+y_(3)+y_(4)=0 and y_(1)y_(2)y_(3)y_(4)=c^(4)`
Hence, all the options are correct.