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If the normals at four points P (xi yi),...

If the normals at four points `P (x_i y_i), i = 1, 2, 3, 4` on the rectangular hyperbola `xy = c^2`, meet at the point Q(h, k), prove that

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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)

If the normal at the point P(x_(i),y_(i)),i=1,2,3,4 on the hyperbola xy=c^(2) are concurrent at the point Q(h,k) then _((x_(1)+x_(2)+x_(3)+x_(4))(y_(1)+y_(2)+y_(3)+y_(4)))((x_(1)+x_(2)+x_(3)+x_(4))(y_(1)+y_(2)+y_(3)+y_(4)))/(x_(1)x_(2)x_(3)x_(4)) is

Knowledge Check

  • 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`

  • 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`

  • If the normals at three points A, B, C on the rectangular hyperbola xy = c^(2) intersect at a . Point D on the hyperbola, then

    A
    circumcircle of `Delta ABC` passes through D
    B
    centroid of triangle ABC lies at the centre of hyperbola `xy = c^(2)`
    C
    orthocentre of `Delta ABC` is the point D
    D
    centre of the circumcircle of `Delta ABC` lies at another hyperbola

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