Question

If $\sec \left(\frac{x+y}{x-y}\right)=a,\mathrm{then}x\mathrm{dy}-y\mathrm{dx}=$

• A. 1
• B. -1
• C. 0.0
• D. a

Answer 1

The correct option is C 0.0

Given that,
$\sec \left(\frac{x+y}{x-y}\right)=a\Rightarrow \frac{x+y}{x-y}=\frac{{\sec }^{-1}a}{1}\mathrm{Applying}\mathrm{Componendo}\mathrm{and}\mathrm{Dividendo}\mathrm{both}\mathrm{sides},\Rightarrow \frac{2x}{2y}=\frac{{\sec }^{-1}a+1}{{\sec }^{-1}a-1}\Rightarrow \frac{y}{x}=\frac{{\sec }^{-1}a-1}{{\sec }^{-1}a+1}\mathrm{Differentiating}\mathrm{both}\mathrm{sides}w.r.t.x,\mathrm{we}\mathrm{get}\Rightarrow \frac{x\cdot \frac{\mathrm{dy}}{\mathrm{dx}}-y\cdot 1}{x^2}=0\Rightarrow x\frac{\mathrm{dy}}{\mathrm{dx}}-y=0\Rightarrow x\mathrm{dy}-y\mathrm{dx}=0$