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Question: Answered & Verified by Expert
If the polar co-ordinates of a point are $\left(\sqrt{2}, \frac{\pi}{4}\right)$, then its Cartesian co-ordinates are
MathematicsStraight LinesMHT CETMHT CET 2021 (21 Sep Shift 1)
Options:
  • A $(\sqrt{2}, 2)$
  • B $(1,-1)$
  • C $(2, \sqrt{2})$
  • D $(1,1)$
Solution:
2514 Upvotes Verified Answer
The correct answer is: $(1,1)$
Polar coordinates $\mathrm{z}=\mathrm{a}+\mathrm{ib}$ are $\left(\sqrt{2}, \frac{\pi}{4}\right)$
$\therefore \sqrt{2}=\sqrt{\mathrm{a}^2+\mathrm{b}^2} \Rightarrow \mathrm{a}^2+\mathrm{b}^2=2$ and $\tan \left(\frac{\pi}{4}\right)=\frac{\mathrm{b}}{\mathrm{a}} \Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=1$
$\Rightarrow \mathrm{a}=\mathrm{b}$
$\therefore \mathrm{a}^2+\mathrm{b}^2=2 \Rightarrow 2 \mathrm{a}^2=2 \Rightarrow \mathrm{a}^2=1 \Rightarrow \mathrm{a}= \pm 1$
Since point lies in $1^{\text {st }}$ quadrant, $a=1 \Rightarrow b=1$
$\therefore$ Cartesian coordinates are $(1,1)$

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