The curve, for which the area of the triangle formed by X -axis, the tangent line at any poin P and line O P is equal to a^2 , is given by

Step-by-step solution

Tangent drawn at any point (x, y) is Y-y= d y d x (X-x) When Y=0, X=x-y d x d y Area of O P Q=a^2 (given) | 1 2 x y |=a^2 array ll & | (x-y d x d y ) y |=2 a^2 & x y-y^2 d x d y = 2 a^2 & d x d y - x y = 2 a^2 y^2 array Here, P=- 1 y and Q= 2 a^2 y^2 aligned IF =e^ P d y & =e^ - 1 y d y & =e^ - y =e^ 1 y = 1 y aligned Hence, required solution is aligned & & x 1 y & = 2 a^2 y^2 1 y d y & & x y & = 2 a^2 y⁻² -2 +C & & x & =C y a^2 y aligned which is the required curve.