Q95 / 150 · MHT CET 2020 (19 Oct Shift 1) · Mathematics

The eccentricity of a rectangular hyperbola is

  • A
    2
  • B
    $2 \sqrt{2}$
  • C
    1
  • Correct answer
    D
    $\sqrt{2}$

Solution

$\lim _{x \rightarrow \pi} \frac{(1+\cos x)-\sin x}{(1+\cos x)+\sin x}$ $=\lim _{x \rightarrow \pi} \frac{2 \cos ^{2} x / 2-2(\sin x / 2) \cos x / 2}{2 \cos ^{2} x / 2+2(\sin x / 2) \cos x / 2}$ $=\lim _{x \rightarrow \pi} \frac{2 \cos ^{2} \frac{x}{2}\left(1-\tan \frac{x}{2}\right)}{2 \cos ^{2} \frac{x}{2}\left(x+\tan \frac{x}{2}\right)}$ $=\lim _{x \rightarrow \pi} \tan \left(\frac{\pi}{4}-\frac{x}{2}\right)=-1$ $\therefore f(\pi)=\lim _{x \rightarrow \pi} f(x)=-1$

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

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