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Question

The vertices of  ABC  lie on a rectangular hyperbola such that the orthocentre of the triangle is (3,2) and the asymptotes of the rectangular hyperbola are parallel to the coordinate axes. If two perpendicular tangents of the hyperbola intersect at the point (1,1) ,theb equation of the rectangular hyperbola is 
  1. xy=2x+y2
  2. None of these
  3. 2xy=x+2y+5
  4. xy=x+y+1


Solution

The correct option is D xy=x+y+1
Let the centre of rectangular hyperbola be(h,k)
then equation of the hyperbola is 
(xh)(yk)=c2

Since perpendicular tangents intersect at the centre of rectangular hyperbola

Hence, centre of hyperbola is (1,1) and equation of the hyperbola will be
(x1)(y1)=c2
for any triangle whose vertices lie on rectangular hyperbola, orthocentre also lie on the same hyperbola
(3,2) lies on the hyperbola
Hence required equation will be
(x1)(y1)=2

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