The area of the triangle formed by the co-ordinate axes and a tangent to the curve x y =a^2 at the point (x₁, y ₁ ) is ________ sq.units (where a, x₁ and y₁ are non zero)

Step-by-step solution

The tangent to the curve xy = a^2 at point (x₁, y₁) has equation y₁x + x₁y = 2a^2 , derived from differentiating xy = a^2 to obtain dy dx = - y x and using point-slope form. The x-intercept occurs at ( 2a^2 y₁ , 0 ) and the y-intercept at (0, 2a^2 x₁ ) , found by setting y = 0 and x = 0 respectively in the tangent equation. The area of the triangle formed by the coordinate axes and this tangent is 1 2 | 2a^2 y₁ | | 2a^2 x₁ | = 4a^4 2|x₁y₁| . Since x₁y₁ = a^2 > 0 , the area simplifies to 2a^4 a^2 = 2a^2 . Final answ