If x2 + y2 =t - 1/t ,x4 + y4 =t2 + 1/t2, prove that xy dy/dx =1 - Maths - Limits and Derivatives

Question

If x2 + y2 =t - 1/t ,x4
+ y4 =t2 + 1/t2, prove that xy
dy/dx =1

Santosh Kumar · Accepted Answer

$x^{2} + y^{2} = t - \frac{1}{t} (1) x^{4} + y^{4} = t^{2} + \frac{1}{t^{2}} \Rightarrow x^{4} + y^{4} = (t - \frac{1}{t})^{2} + 2 [Using equation (1)] \Rightarrow x^{4} + y^{4} = (x^{2} + y^{2})^{2} + 2 \Rightarrow x^{4} + y^{4} = x^{4} + y^{4} + 2 x^{2} y^{2} + 2 \Rightarrow 2 (x^{2} y^{2} + 1) = 0 \Rightarrow x^{2} y^{2} + 1 = 0$ Differentiating both sides with respect to x, $x^{2} \frac{d}{d y} (y^{2}) + y^{2} \frac{d}{d x} (x^{2}) = 0 \Rightarrow 2 x^{2} y \frac{d y}{d x} + 2 y^{2} x = 0 \Rightarrow 2 x y (x \frac{d y}{d x} + y) = 0 \Rightarrow x \frac{d y}{d x} + y = 0 \Rightarrow - \frac{x}{y} \frac{d y}{d x} = 1$