Question

If $xy^{2}=ax^{2}+bxy+y^{2}$, then find $\frac{dy}{dx}$.

• A. $\frac{2ax+by+y^{2}}{2xy+bx+2y}$
• B. $\frac{2ax+by-y^{2}}{2xy-bx-2y}$
• C. $\frac{ax+by+xy}{xy+x^{2}+y^{2}}$
• D. $\frac{2x^{2}+axy+y^{2}}{x^{2}+y^{2}+2xy}$

Answer 1

Answer: (B)

We have, $xy^2 = ax^2 + bxy + y^2$
Differentiating $w$.$r$.$t$. $x$, we get
$x\cdot2y \frac{dy}{dx}+y^{2}=2ax+b\left(x \frac{dy}{dx}+y\right)+2y \frac{dy}{dx}$
$\Rightarrow \left(2xy-bx-2y\right) \frac{dy}{dx}$
$=2ax+by-y^{2}$
$\Rightarrow \frac{dy}{dx}=\frac{2ax+by-y^{2}}{2xy-bx-2y}$