Question

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`

A. `x_(1)+x_(2)+x_(3) +x_(4)=0`
B. `y_(1)+y_(2)+y_(3) +y_(4)=0`
C. `x_(1)x_(2)x_(3)x_(4)=c^(4)`
D. `y_(1)y_(2)y_(3)y_(4)=c^(4)`

Answer 1

It is given that,
`x^(2)+y^(2)=a^(2) " …(i)" `
and ` xy=c^(2) " …(ii)" `
We obtain ` x^(2)+c^(4)//x^(2)=a^(2)`
`rArr x^(4)-a^(2)x^(2)+c^(4)=0 " …(iii)" `
Now `x_(1),x_(2),x_(3),x_(4)` will be root of Eq. (iii).
Therefore, `Sigma x_(1)=x_(1)+x_(2)+x_(3)+x_(4)=0`
and product of the roots `x_(1)x_(2)x_(3)x_(4)=c^(4)`
Similarly, `y_(1)+y_(2)+y_(3)+y_(4)=0`
and ` y_(1)y_(2)y_(3)y_(4)=c^(4)`
Hence, all options are correct.