Let the mid-point be (h,k).
Equation of chord with mid point (h,k) is given by :
T = S1
xh/a2 - yk/b2 - 1 = h2/a2 - k2/b2 - 1
Make this equation of chord homogeneous with the curve x2/a2 - y2/b2 = 1
i.e x2/a2-y2/b2=(xh/a2 - yk/b2)2/(h2/a2 - k2/b2 )2
1/a2(h2/a2 - k2/b2 )2x2-1/b2(h2/a2 - k2/b2 )2 y2 = h2/a4 x2+ k2/b4 y2-2hk/a2b2
the lines are perpendicular if coefficient of x2 + coefficient of y2 = 0
i.e 1/a2(h2/a2 - k2/b2 )2 -h2/a4 -1/b2(h2/a2 - k2/b2 )2 - k2/b4 =0
(h2/a2 - k2/b2 )2(1/a2-1/b2)=h2/a4 +k2/b4
hence the locus by putting x=h and y=k
Moderation Team
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