If x = a ( θ – sin θ ), y = a (1 + cos θ ), find d^ 2 y dx^ 2 . - Vidyadip

Question

If x = a ( θ – sin θ ), y = a (1 + cos θ ), find $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$.

✓

Answer

$\therefore\frac{\text{dx}}{\text{d}\theta}=\text{a(1 - cos}\theta)\text{ and }\frac{\text{dy}}{d\theta}=-\text{a sin}\theta$$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{-\sin\theta}{(1-\cos\theta)}$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\frac{\text{(1-cos}\theta)\text{(-cos}\theta)+\sin\theta(\sin\theta)}{(1-\cos\theta)^{2}}\cdot\frac{\text{d}\theta}{\text{dx}}$
= $\frac{(\text{1-cos}\theta)}{\text{(1 - cos}\theta)^{2}}\cdot\frac{1}{\text{a(1 - cos}\theta)}$
$= \frac{1}{\text{a(1- cos}\theta)^{2}}\text{ or }\frac{1}{\text{4a}}\text{cosec}^{4}\frac{\theta}{2}$

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