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GATE 2021 BT – Question 15

Engineering Mathematics · Calculus: Limits, continuity, differentiability, partial derivatives, maxima and minima · 1 mark · Multiple choice

The Cartesian coordinates of a point with polar coordinates $(4,\pi/4)$ are

  1. $(\sqrt3,2\sqrt2)$
  2. $(2,2\sqrt3)$
  3. $(2\sqrt2,\sqrt3)$
  4. $(2\sqrt2,2\sqrt2)$

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Show answer and explanation

Correct answer: (D) $(2\sqrt2,2\sqrt2)$

Explanation

To convert polar coordinates $(r,\theta)$ to Cartesian coordinates $(x,y)$ use
$$x=r\cos\theta,\qquad y=r\sin\theta .$$

With $r=4$ and $\theta=\pi/4$ (so $\cos\theta=\sin\theta=\dfrac1{\sqrt2}$):
$$x=4\cdot\frac1{\sqrt2}=2\sqrt2,\qquad y=4\cdot\frac1{\sqrt2}=2\sqrt2 .$$

The point is $(2\sqrt2,\ 2\sqrt2)$ (option **D**). Check: $x^2+y^2=8+8=16=r^2$ ✓.