Answer: Option B
:
B
Let the equation to the rectangular hyperbola be
$x^2-y^2=a^2$ …… (i)
As the asymptotes of this are the axes of the other and vice-versa, hence the equation of the other hyperbola may be written as
$xy=c^2$ …… (ii)
Let equations (i) and (ii) meet at some point whose coordinates are
$(asec\alpha ,atan\alpha )$
Then, the tangent at the point a $(sec\alpha ,atan\alpha )$ to equation (ii) is
$x-ysin\alpha =acos\alpha$ …… (iii)
and the tangent at the point $(asec\alpha ,atan\alpha )$ to equation (ii) is
$y+xsin\alpha =\left(\frac{2c^2}{a}\right)cos\alpha$ …… (iv)
clearly, the slopes of the tangents given by equations (iv) and (iii) are respectively - $sec\alpha and\frac{1}{sin\alpha }$ , so their product is $-sec\alpha .\left(\frac{1}{sin\alpha }\right)=-1$ Hence, the tangents are at right angle.