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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability2 Marks
Question
If $x$ and $y$ are connected parametrically by the equation $x = 2at^2, y = at^4,$ without eliminating the parameter, Find$\frac{{dy}}{{dx}}.$
✓

Answer

Given: $x = 2at^2$ and $y = at^4$
$\therefore \frac{{dx}}{{dt}} = \frac{d}{{dt}}\left( {2a{t^2}} \right)$ and $\frac{{dy}}{{dt}} = \frac{d}{{dt}}\left( {a{t^4}} \right)$
$ \Rightarrow \frac{{dx}}{{dt}} = 2a\frac{d}{{dt}}\left( {{t^2}} \right) = 2a.2t = 4at$ and $\frac{{dy}}{{dt}} = a\frac{d}{{dt}}\left( {{t^4}} \right) = a.4{t^3} = 4a{t^3}$
Now $\frac{{dy}}{{dx}} = \frac{{dy/dt}}{{dx/dt}} = \frac{{4a{t^3}}}{{4at}} = {t^2}$

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