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Question

Tangent at any point on the hyperbola x2a2y2b2=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


  1. None of these


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.

p (x coordinate of A, y coordinate of B) (h. k)

Then      A(h,0)

             B(0,k)

The given hyperbola is,

x2a2y2b2=1

Whose tangent ig given by,

y=mx±a2m2b2

This passes through (h,0)

0=mh±a2m2b2

mh=a2m2b2

m2h2=a2m2b2

m2=b2a2h2          - - - - - - -(1)

Tangent also passes through (0,k)

k=a2m2b2

k2+b2a2=m2          - - - - - -(2)

(1) and (2)

b2a2h2=k2+b2a2=m2 

a2b2=a2k2+a2b2h2k2h2b2

dividing throughout by h2k2 

o=a2h21b2k2

i.e.,a2h2b2k2

Since (h,k) gives point P,the locus can be given as

a2x2b2y2=1

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