If {\;y/x}+{\;x/y}=6, then dy/dx is

Correct Answer is : x−17y17x−yx-17y/17x-y17x−yx−17y

Correct Answer is : x−17y17x−yx-17y/17x-y17x−yx−17y

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Differential Calculus

Question : 35 of 69

Marks: +1, -0

\sqrt{\;\frac{y}{x}}+\sqrt{\;\frac{x}{y}}=6,

\frac{dy}{dx}

Solution:

Concept:

Use substitution and implicit differentiation to find

\frac{dy}{dx}

Explanation:

Let

t = \sqrt{\frac{y}{x}}

, then

\sqrt{\frac{x}{y}} = \frac{1}{t}

Given:

t + \frac{1}{t} = 6

. Multiplying by

t

t^2 - 6t + 1 = 0

Alternatively, multiply the original equation by

\sqrt{xy}

x + y = 6\sqrt{xy}

Square both sides:

(x+y)^2 = 36xy

⇒

x^2 + 2xy + y^2 = 36xy

⇒

x^2 - 34xy + y^2 = 0

Differentiate implicitly w.r.t.

x

2x - 34\left(y + x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0

Simplify:

2x - 34y - 34x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0

Collect

\frac{dy}{dx}

terms:

( -34x + 2y)\frac{dy}{dx} + (2x - 34y) = 0

Thus

\frac{dy}{dx} = \frac{34y - 2x}{-34x + 2y} = \frac{17y - x}{y - 17x} = \frac{x - 17y}{17x - y}

Answer:

\frac{dy}{dx} = \frac{x - 17y}{17x - y}

, which corresponds to option C.

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