Q65 / 160 · TS EAMCET 2024 (11 May Shift 1) · Mathematics

If $2 \tan ^2 \theta-4 \sec \theta+3=0$, then $2 \sec \theta=$

  • A
    $3$
  • B
    $2+\sqrt{2}$ and $2-\sqrt{2}$
  • C
    $2-\sqrt{2}$
  • Correct answer
    D
    $2+\sqrt{2}$

Solution

$\begin{aligned} & 2 \tan ^2 \theta-4 \sec \theta+3=0 \\ & \Rightarrow 2 \sec ^2 \theta-4 \sec \theta+1=0 \\ & \Rightarrow \sec \theta=\frac{4 \pm \sqrt{16-8}}{4} \Rightarrow \sec \theta=\frac{2 \pm \sqrt{2}}{2} \\ &|\sec \theta|\gt1 \Rightarrow \sec \theta \neq \frac{2-\sqrt{2}}{2} \Rightarrow 2 \sec \theta=2+\sqrt{2} .\end{aligned}$

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

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