What is rac{d²y}{dx² equal to ? [NDA I 2015]

Correct Answer is : −a2y3-a²/y³−y3a2

Correct Answer is : −a2y3-a²/y³−y3a2

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UPSC NDA I 2015 Mathematics Solved Paper

For the next three (3) items that follow :

x=\frac{a(1-t^2)}{1+t^2},y=\frac{2at}{1+t^2}

[NDA I 2015]

Question : 84 of 120

Marks: +1, -0

What is

rac{d^2y}{dx^2}

Solution:

Differentiating the equation of the circle obtained in the previous question yields

x^2+y^2=a^2

Differentiating with respect to x

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

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

Taking the second derivative gives us

rac{d^2y}{dx^2}

=\frac{d(y')}{dx}

=\frac{d}{dx}\left(\frac{-x}{y}\right)

=\frac{y(-1)+x(y')}{y^2}

= \frac{xy' - y}{y^2}

=\frac{x}{y^2}\frac{dy}{dx}-\frac{1}{y}

=\frac{x(-x)}{y^3}-\frac{1}{y}

=\frac{-x^2-y^2}{y^3}

=\frac{-a^2}{y^3}

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