question_answer

The area of the triangle formed by the tangent to the hyperbola \[xy={{a}^{2}}\] and co-ordinate axes is              [RPET 2000]

A)            \[{{a}^{2}}\]                         

B)            \[2{{a}^{2}}\]

C)            \[3{{a}^{2}}\]                       

D)            \[4{{a}^{2}}\]

Correct Answer: B

Solution :

           Given \[xy={{a}^{2}}\] or \[y=\frac{{{a}^{2}}}{x}\]                 .....(i)                There are two points on the curve (a, a),(? a,? a)            The equation of the line at \[(a,a)\]is,            \[y-a={{\left( \frac{dy}{dx} \right)}_{(a,\,a)}}(x-a)\]\[={{\left( \frac{-{{a}^{2}}}{{{x}^{2}}} \right)}_{(a,\,a)}}(x-a)\]            \[y-a=-(x-a)\] therefore, equation of the tangent at \[(a,a)\] is \[x+y=2\,a\].The interception of line \[x+y=2a\] with x-axis is 2a and with y-axis is 2a.            \[\therefore \] Required area = \[\frac{1}{2}\times 2a\times 2a=2{{a}^{2}}\].