Given the parametric equations x-ft-y-gt- Then d 2 y - d x 2 equals

The correct option is B d ^ 2 y d t ^ 2 . dx dt dy dt . d ^ 2 x d t ^ 2 #160;( dx dt #160;) #160;^ 3 We have dy dx = dy dt ...

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Standard XII

Mathematics

Parametric Differentiation

Given the par...

Question

Given the parametric equations $x = f (t), y = g (t) .$ Then $\frac{d^{2} y}{d x^{2}}$ equals

\frac{\frac{d^{2} y}{d t^{2}} . \frac{d x}{d t} - \frac{d y}{d t} . \frac{d^{2} x}{d t^{2}}}{{(\frac{d x}{d t})}^{2}}

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\frac{\frac{d^{2} y}{d t^{2}} . \frac{d x}{d t} - \frac{d y}{d t} . \frac{d^{2} x}{d t^{2}}}{{(\frac{d x}{d t})}^{3}}

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\frac{\frac{d^{2} y}{d t^{2}}}{\frac{d^{2} x}{d t^{2}}}

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Solution

The correct option is B $\frac{\frac{d^{2} y}{d t^{2}} . \frac{d x}{d t} - \frac{d y}{d t} . \frac{d^{2} x}{d t^{2}}}{{(\frac{d x}{d t})}^{3}}$
We have $\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
Again differentiating both sides w.r.t x,
we get $\frac{d^{2} y}{{d x}^{2}} = \frac{d}{d t} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{\frac{d y}{d t}}{\frac{d x}{d t}} ⎞ ⎟ ⎟ ⎟ ⎠ \frac{d t}{d x}$
$= \frac{\frac{d^{2} y}{{d t}^{2}} (\frac{d x}{d t}) - (\frac{d^{2} x}{{d t}^{2}}) \frac{d y}{d t}}{{(\frac{d t}{d t})}^{2}} \times \frac{1}{\frac{d x}{d t}}$
$= \frac{\frac{d^{2} y}{{d t}^{2}} . (\frac{d x}{d t}) - (\frac{d^{2} x}{{d t}^{2}}) . (\frac{d y}{d t})}{{(\frac{d x}{d t})}^{3}}$

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