141.

$\mathrm{If} {\cos }^{-1}\left(\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) = \mathrm{k} \left(\mathrm{a} \mathrm{constant}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}$

• $\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}$

$$\begin{gathered} \mathrm{Given}, {\cos }^{-1}\left(\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) = \mathrm{k} \\ \frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2} = \cos \left(\mathrm{k}\right) \\ \mathrm{On} \mathrm{differentiating} \mathrm{w}.\mathrm{r}.\mathrm{t}. \mathrm{x}, \mathrm{we} \mathrm{get} \\ \frac{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)\left(2\mathrm{x} - 2\mathrm{y}{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right) - \left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\left(2\mathrm{x} + 2\mathrm{y}{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right) }{{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)}^2} = 0 \\ \Rightarrow - 4{\mathrm{x}}^2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} + 4{\mathrm{xy}}^2 = 0 \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}}{\mathrm{x}} \end{gathered}$$