The area of a triangle formed by a tangent to the curve `2xy =a^(2)` and the coordinate axes, isA. `2a^(2)`B. `a^(2)`C. `3a^(2)`D. none of these

The area of a triangle formed by a tangent to the curve `2xy =a^(2)` a

1.	The area of a triangle formed by a tangent to the curve `2xy =a^(2)` and the coordinate axes, isA. `2a^(2)`B. `a^(2)`C. `3a^(2)`D. none of these
Answer» Correct Answer - B 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)]

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The area of a triangle formed by a tangent to the curve `2xy =a^(2)` and the coordinate axes, isA. `2a^(2)`B. `a^(2)`C. `3a^(2)`D. none of these

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