Q88 / 180 · KCET 2009 · Mathematics

The length of the subtangent at $t$ on the curve $\mathrm{x}=\mathrm{a}(\mathrm{t}+\sin \mathrm{t}), \mathrm{y}=\mathrm{a}(1-\cos \mathrm{t})$ is

  • Correct answer
    A
    $a \sin \mathrm{t}$
  • B
    $2 \mathrm{a} \sin \left(\frac{\mathrm{t}}{2}\right) \tan \left(\frac{\mathrm{t}}{2}\right)$
  • C
    $2 \mathrm{a} \sin \frac{\mathrm{t}}{2}$
  • D
    $2 a \sin ^{3}\left(\frac{t}{2}\right) \sec \left(\frac{t}{2}\right)$

Solution

Given, $x=a(t+\sin t), y=a(1-\cos t)$ $$ \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{a}(1+\cos \mathrm{t}), \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{a}(\sin \mathrm{t}) $$ $$ \therefore \quad \frac{d y}{d x}=\frac{a \sin t}{a(1+\cos t)}=\tan \frac{t}{2} $$ $$ \begin{aligned} \therefore \text { Length of subtangent } &=\frac{y}{d y / d x} \\ &=\frac{a(1-\cos t)}{\tan \frac{t}{2}} \\ &=2 a \sin \frac{t}{2} \cos \frac{t}{2} \\ &=a \sin t \end{aligned} $$

Hence, the correct answer is option: 'A'.

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