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any point on a rectangular hyperbola xy=c^{2} is P(ct,c/t)..and slope at p is given by -1/t^{2} . write the equation of line(which is tangent to the given hyperbola) with slope -1/t^{2} and passing throught P...

then write the above equation of line in the form of x/a + y/b =1... you'll get a=2ct and b=2c/t. now the are of triangle is ab/2 which is 2c^{2.. } given c^{2}=2 so are of triangle is 4

NOTE: The asymptotes of rectangular hyperbola xy=c^{2} are the coordinate axes.