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If the circle x^(2)+y^(2)=a^(2) intersec...

If the circle `x^(2)+y^(2)=a^(2)` intersects the hyperbola `xy=c^(2)` at four points `P(x_(1),y_(1)), Q(x_(2),y_(2)), R(x_(3),y_(3)), and S(x_(4),y_(4))`, then

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)`

Text Solution

Verified by Experts

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.
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