If x = a t^2 ,y = 2at, then d^2 y d x^2 = - Vidyadip

MCQ

If $x = a{t^2},y = 2at$, then ${{{d^2}y} \over {d{x^2}}} = $

A
$ - {1 \over {{t^2}}}$
B
${1 \over {2a{t^3}}}$
C
$ - {1 \over {{t^3}}}$
✓
$ - {1 \over {2a{t^3}}}$

✓

Answer

Correct option: D.

$ - {1 \over {2a{t^3}}}$

d
(d) $\frac{{dy}}{{dx}} = \frac{{dy/dt}}{{dx/dt}} = \frac{{2a}}{{2at}}$

==> $\frac{{dy}}{{dx}} = \frac{1}{t} = \frac{{2a}}{y}$

==> $y\frac{{dy}}{{dx}} = 2a$

==> $y\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^2} = 0$

$ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = \frac{{ - {{(dy/dx)}^2}}}{y} = - \frac{1}{{2a{t^3}}}$.

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