If x = a ( 2t +2t 2t) and y = a ( 2t - 2t 2t),then find d^ 2 y dx… - Vidyadip

Question

$\text{If x = a} (\cos 2\text{t +2t}\sin \text{2t}) \text{and y = a} (\sin \text{2t - 2t}\cos\text{2t}),\text{then find}\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}}.$

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Answer

$\text{x = a} (\cos \text{2t + 2t} \sin \text{2t)}$
$\text{y = a} (\sin \text{2t - 2t} \cos \text{2t)}$
$\Rightarrow \frac{\text{dx}}{\text{dt}} = 4\text{ at} \cos \text{2 t}$
$\Rightarrow \frac{\text{dy}}{\text{dt}} = 4\text{ at} \sin \text{2 t}$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = \tan \text{2 t}$
$\Rightarrow \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = 2 \sec^{2} \text{2 t}. \frac{\text{dt}}{\text{dx}}$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = \frac{1}{2 \text{at} \cos^{3} \text{2t}}$

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