If the normal at four points `P_(i)(x_(i), (y_(i)) l, I = 1, 2, 3, 4` on the rectangular hyperbola `xy = c^(2)` meet at the point `Q(h, k),` prove that
`x_(1) + x_(2) + x_(3) + x_(4) = h, y_(1) + y_(2) + y_(3) + y_(4) = k`
`x_(1)x_(2)x_(3)x_(4) =y_(1)y_(2)y_(3)y_(4) =-c^(4)`

Options

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

Answer

A, B, D

Explanation

Any point on `xy=2` is `P(sqrt(2)t,(sqrt(2))/t)`
Normal at `P` is `y-(sqrt(2))/t=t^(2)(x-sqrt(2)t)`
`4t-sqrt(2)=t^(3)-sqrt(2)t`
`sqrt(2)t^(4)-3t^(3)+4t-sqrt(2)=0`
`t_(1)+t_(2)+t_(3)+t_(4)=3/(sqrt(2))impliesx_(1)+x_(2)+x_(3)+x_(4)=3`
`t_(1)t_(2)t_(3)t_(4)=-12sqrt(2)implies1/(t_(1))+1/(t_(2))+1/(t_(3))+1/(t_(4))=2sqrt(2)`
`impliesy_(1)+y_(2)+y_(3)+y_(4)=4`
`y_(1)y_(2)y_(3)y_(4)=4/(t_(1)t_(2)t_(3)t_(4))=-4`