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Hi Mayur,
Consider the point P(3sect,2tant) be any point on the hyperbola. (Refer the diagram, you'd asked for):
So the equation of the chord of contact from this point is (T=0)... which is
x(3sect) + y(2tant) = 9 ----------(1)
Now the equation of the chord of contact, in terms of it's midpoint M(h.k) is (T=S1) which is,
x(h) + y(k) = h2 + k2 ----------(2)
Both (1) and (2) represent the same straight line,
So 3sect/h = 2tant/k = 9/(h2 + k2)
to find the relation interms of h,k try to eliminate t from the above relation.
So, sect = 3h/(h2 + k2) and tant = 9k/2(h2 + k2).
And we know, sec2t - tan2t = 1,
So 9h2 - (9k/2)2 = (h2 + k2)2.
And hence the locus is 9x2 - (9y/2)2 = (x2+y2)2
Hope that helps.
Best Regards,
Ashwin (IIT Madras).
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