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Question

Tangent at any point on the hyperbola x2a2âˆ’y2b2=1 cut the axis at A and B respectively. If the rectangle OAPB (where O is origin) is completed then locus of point P is given by

A

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B

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C

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D

None of these

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Solution

The correct option is A Lets draw the diagram with given hyperbola and its tangent Here APBO forms a rectangle. Since A and B represents point on x And y axis. pâ‰¡ (x coordinate of A, y coordinate of B) â‰¡ (h. k) Then Aâ‰¡(h,0) Bâ‰¡(0,k) The given hyperbola is, x2a2âˆ’y2b2=1 Whose tangent ig given by, y=mxÂ±âˆša2m2âˆ’b2 This passes through (h,0) 0=mhÂ±âˆša2m2âˆ’b2 mh=âˆša2m2âˆ’b2 m2h2=a2m2âˆ’b2 m2=b2a2âˆ’h2 - - - - - - -(1) Tangent also passes through (0,k) k=âˆša2m2âˆ’b2 k2+b2a2=m2 - - - - - -(2) (1) and (2) â‡’ b2a2âˆ’h2=k2+b2a2=m2 a2b2=a2k2+a2b2âˆ’h2k2âˆ’h2b2 dividing throughout by h2k2 o=a2h2âˆ’1âˆ’b2k2 i.e.,a2h2âˆ’b2k2 Since (h,k) gives point P,the locus can be given as a2x2âˆ’b2y2=1

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