GATE 2018 EC Q.12: solution and answer

GATE 2018 · ECQ.12 · 1 mark · MCQ

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

(A)
$a = \frac{b}{4}$
(B)
$a = \frac{b}{2}$
(C)
$a = 2 b$
(D)
$a = 4 b$

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The condition gives $a = 4 b$ , which is option (D).

Official answer: DSource: Official answer key, GATE 2018 EC (IIT Guwahati)

Split the fraction before differentiating: $f (x, y) = \frac{a x ^{2} + b y ^{2}}{x y} = a \frac{x}{y} + b \frac{y}{x}$ .
Differentiate with respect to $x$ , holding $y$ constant: $\frac{\partial f}{\partial x} = \frac{a}{y} - \frac{b y}{x ^{2}}$ .
Differentiate with respect to $y$ , holding $x$ constant: $\frac{\partial f}{\partial y} = - \frac{a x}{y ^{2}} + \frac{b}{x}$ .
Substitute $x = 1, y = 2$ : $\frac{\partial f}{\partial x} = \frac{a}{2} - 2 b$ and $\frac{\partial f}{\partial y} = - \frac{a}{4} + b$ .
Set the two equal: $\frac{a}{2} - 2 b = - \frac{a}{4} + b \Rightarrow \frac{3 a}{4} = 3 b \Rightarrow a = 4 b$
Check: take $b = 1, a = 4$ . Then $\frac{\partial f}{\partial x} = 2 - 2 = 0$ and $\frac{\partial f}{\partial y} = - 1 + 1 = 0$ , so the two are equal. The answer is option (D).

KEY IDEA

Write $f$ as $a \frac{x}{y} + b \frac{y}{x}$ first. Each term is then a simple power of $x$ or $y$ , so both partial derivatives take one line each and you never need the quotient rule. Put in the point, set the two derivatives equal and solve the linear equation for $a$ in terms of $b$ .

WATCH OUT FOR

The usual slip is a sign or power when using the quotient rule on the whole fraction. For example, writing $\frac{\partial}{\partial x} (\frac{b y}{x}) = \frac{b y}{x ^{2}}$ without the minus sign gives $\frac{a}{2} + 2 b = - \frac{a}{4} + b$ , which leads to $a = - \frac{4 b}{3}$ , and that matches no option. Swapping which variable is held constant (treating $\frac{a x}{y}$ as $a$ times $\frac{1}{y}$ when differentiating in $x$ ) produces the distractors $a = \frac{b}{4}$ or $a = \frac{b}{2}$ .

From the GATE 2018 EC paper. Every question of this paper · Download the full paper · All 39 Calculus questions, with notes

GATE 2018 EC Q.12: solution and answer

Answer and solution

More Calculus questions