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  • If <img alt="\mathrm{\cos ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\log a}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Ccos%20%5E%7B-1%7D%5Cleft%28%5Cfrac%7Bx%5E%7B

If \mathrm{\cos ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\log a} then \mathrm{\frac{\mathrm{d} y}{\mathrm{~d} x}} is equal to

Option: 1

\mathrm{y / x}


Option: 2

\mathrm{x / y}


Option: 3

\mathrm{x^{2} / y^{2}


Option: 4

\mathrm{y^{2}/x^{2}}.


Answers (1)

best_answer

\mathrm{\cos ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\log a \Rightarrow \frac{x^{2}-y^{2}}{x^{2}+y^{2}}=\cos \log a=A (say)}

Putting \mathrm{u=y / x} and applying componendo and dividendo, we have

\begin{aligned} & (y / x)^{2}=u^{2}=(1-A) /(1+A) \\ & \Rightarrow y / x=\sqrt{(1-A) /(1+A)} \\ & \Rightarrow \quad x \mathrm{~d} y / \mathrm{d} x-y=0 \\ & \Rightarrow \mathrm{d} y / \mathrm{d} x=y / x \end{aligned}

Posted by

Sanket Gandhi

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