Q80 / 180 · KCET 2015 · Mathematics

If \( x=a \cos ^{3} \theta, y=a \sin ^{3} \theta \), then \( 1+\left(\frac{d y}{d x}\right)^{2} \) is

  • A
    \( \tan \theta \)
  • B
    \( \tan ^{2} \theta \)
  • Correct answer
    C
    \( \sec ^{2} \theta \)
  • D
    \( 11 \)

Solution

Given that $x=a \cos ^{3} \theta$ and $y=a \sin ^{3} \theta$ Now, $\frac{d x}{d \theta}=a\left(3 \cos ^{2} \theta\right)(-\sin \theta)$ and $\frac{d y}{d \theta}=a\left(3 \sin ^{2} \theta\right)(\cos \theta)$ Thus, $\frac{d y}{d x}=\frac{d y}{d \theta} \times \frac{d \theta}{d x}$ $=\frac{3 a \sin ^{2} \theta \cos \theta}{-3 a \cos ^{2} \theta \sin \theta}=\frac{-\sin \theta}{\cos \theta}$ $\Rightarrow \frac{d y}{d x}=-\tan \theta \rightarrow(1)$ So, $1+\left(\frac{d y}{d x}\right)^{2}=1+\left(-\tan ^{2} \theta\right)$ $=1+\tan ^{2} \theta=\sec ^{2} \theta$

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

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