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If the normals at `(x_(i),y_(i)) i=1,2,3,4` to the rectangular hyperbola `xy=2` meet at the point `(3,4)` then(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`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` |
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Answer» Correct Answer - A::B::D 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` |
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