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GATE 2018 EC – Question 22

Engineering Mathematics · Calculus · 1 mark · Multiple choice

Let $f(x,y)=\dfrac{ax^2+by^2}{xy}$, where $a$ and $b$ are constants. If $\dfrac{\partial f}{\partial x}=\dfrac{\partial f}{\partial y}$ at $x=1$ and $y=2$, then the relation between $a$ and $b$ is

  1. $a=\dfrac b4$
  2. $a=\dfrac b2$
  3. $a=2b$
  4. $a=4b$

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Correct answer: (D) $a=4b$

Explanation

$f=\frac{ax}{y}+\frac{by}{x}$, so $f_x=\frac ay-\frac{by}{x^2}$ and $f_y=-\frac{ax}{y^2}+\frac bx$. At $(1,2)$: $\frac a2-2b=-\frac a4+b$, so $\frac{3a}{4}=3b$ and $a=4b$.