If the normal at `(x_(i) y_(i))` i=1,2,3,4 on `xy=c^2` meet at the point `(alpha, beta)` show that `sum x_(i)=alpha, sum y_(i)=beta, sum x_(i)^(2)=alpha^(2), sum y^(2)=beta^(2), x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=-c^(4)`

A circle cuts the rectangular hyperbola xy=1 in the points (x_(1),y_(1)), r=1,2,3,4 . Prove that x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=1

If the normals at (x_(1),y_(1)) and (x_(2) ,y_(2)) on y^(2) = 4ax meet again on parabola then x_(1)x_(2)+y_(1)y_(2)=

If the hyperbola xy=c^(2) intersects the circle x^(2)+y^(2)=a^(2)" is four points "P(x_(1),y_(1)), Q(x_(2),y_(2)), R(x_(3),y_(3)) and S(x_(4),y_(4)) then show that (i) x_(1)+x_(2)+x_(3)+x_(4)=0 (ii) y_(1)+y_(2)+y_(3)+y_(4)=0 (iii) x_(1)x_(2)x_(3)x_(4)=c^(4) (iv) y_(1)y_(2)y_(3)y_(4)=c^(4)

If a circle cuts the rectangular hyperbola xy = 1 in the points (x_(r), y_(r)), r = 1, 2, 3, 4 prove that x_(1), x_(2),x_(3), x_(4) = y_(1), y_(2), y_(3), y_(4) = 1

If (x_i y_i) ,i =1,2,3,4 are concylic points on xy= 4 then x_1x_2x_3x_4 =

If alpha , beta , gamma are the roots of 2x^3 -2x -1=0 the sum ( alpha beta)^2 =

If alpha, beta, gamma, delta are roots of x^(3) - 4x^(2) - x + 2 = 0 then sum (1)/(alpha^(2)) =

If alpha, beta, gamma are roots of x^(3) - 2x^(2) + 3x - 4 = 0 , then sum a^(2) beta^(2) =

If alpha , beta , gamma are the roots of the equation x^3 +4x^2 -5x +3=0 then sum (1)/( alpha^2 beta^2)=