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AP DSC School Assistant Math Model Test 1

Section: Subject Specific

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Question : 60 of 160

Marks: +1, -0

If

x=a(t+\frac{1}{t})

y=a(t-\frac{1}{t})

\frac{dx}{dy}

is:

$\frac{x}{y}$
$\frac{y}{x}$
$\frac{1}{x}$
1

Solution:

Concept:

Parametric differentiation: find

\frac{dx}{dy}

by computing

\frac{dx/dt}{dy/dt}

using the derivative of power functions.

Explanation:

Given

x = a\left(t + \frac{1}{t}\right)

and

y = a\left(t - \frac{1}{t}\right)

.

Differentiate both with respect to

t

:

\frac{dx}{dt} = a\left(1 - \frac{1}{t^{2}}\right)

\frac{dy}{dt} = a\left(1 + \frac{1}{t^{2}}\right)

Now

\frac{dx}{dy} = \frac{dx/dt}{dy/dt} = \frac{a\left(1 - \frac{1}{t^{2}}\right)}{a\left(1 + \frac{1}{t^{2}}\right)} = \frac{1 - \frac{1}{t^{2}}}{1 + \frac{1}{t^{2}}}

.

Multiply numerator and denominator by

t

:

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

Now note that

t - \frac{1}{t} = \frac{y}{a}

and

t + \frac{1}{t} = \frac{x}{a}

, so

\frac{dx}{dy} = \frac{y/a}{x/a} = \frac{y}{x}

.

Answer:

\frac{y}{x}

(Option B)

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