JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023If 2xy+3yx=20, then dxdy at (2,2) is equal to :
Choose the correct answer:
- A.
−(2+loge43+loge8)
−(3+loge42+loge8)
Explanation
\begin{aligned}
& 2x^{y}+3y^{x}=20 \\
& \Rightarrow2x^{y}\left(y.\frac{1}{x}+\ln x\frac{dy}{dx}\right)+3y^{x}\left(x\frac{1}{y}\frac{dy}{dx}+\ln y.1\right)=0 \\
& \mathrm{but}\left(2,2\right) \\
& \Rightarrow2.2^{2}\left(1+\ln2.\frac{dy}{dx}\right)+3.4\left(1\frac{dy}{dx}+\ln2\right)=0 \\
& \Rightarrow\frac{dy}{dx}[8\ln2+12]+8+12\ln2=0 \\
& \text{一}\frac{dy}{dx}=-\left[\frac{2+\ln8}{3+\ln4}\right]
\end{aligned}
Explanation
\begin{aligned}
& 2x^{y}+3y^{x}=20 \\
& \Rightarrow2x^{y}\left(y.\frac{1}{x}+\ln x\frac{dy}{dx}\right)+3y^{x}\left(x\frac{1}{y}\frac{dy}{dx}+\ln y.1\right)=0 \\
& \mathrm{but}\left(2,2\right) \\
& \Rightarrow2.2^{2}\left(1+\ln2.\frac{dy}{dx}\right)+3.4\left(1\frac{dy}{dx}+\ln2\right)=0 \\
& \Rightarrow\frac{dy}{dx}[8\ln2+12]+8+12\ln2=0 \\
& \text{一}\frac{dy}{dx}=-\left[\frac{2+\ln8}{3+\ln4}\right]
\end{aligned}