The area of a triangle formed by a tangent to the curve `2xy =a^(2)` and the coordinate axes, is

Options

A. `2a^(2)`
B. `a^(2)`
C. `3a^(2)`
D. none of these

Answer

B

Explanation

Let `P(x_(1), y_(1))` be a point on the curve
`2xy =a^(2) " " `…(i)
Then,
`2x_(1)y_(1)=a^(2) " " `…(ii)
Now,
`2xy=a^(2) rArr 2(x(dy)/(dx)+y) =0 rArr (dy)/(dx)= -(y)/(x) rArr ((dy)/(dx))_(p) = -(y_(1))/(x_(1))`
The equation of the tangent to (i) at `P(x_(1),y_(1))` is
`y-y_(1)=-(y_(1))/(x_(1))(x-x_(1))`
`rArr x_(1)y-x_(1)y_(1)= -xy_(1) + x_(1)y_(1)`
`rArr xy_(1)+yx_(1)=2x_(1)y_(1) rArr xy_(1)+yx_(1)=a^(2) " " `[Using (ii)]
This tangent meets the coordinate axes at
`A(a^(2)//y_(1),0) and B(0, a^(2)//x_(1))`
`therefore " Area of " Delta OAB = (1)/(2) xx OA xx OB`
`rArr " Area of " Delta OAB=(1)/(2) xx (a^(2))/(y_(1)) xx (a^(2))/(x_(1))=(a^(4))/(2x_(1)y_(1))=a^(2) " " `[Using (i)]