Question

A line passes through the points \(O(0,0),A(3,3)\). Then the polar coordinates of point A are ...

• A. \(\left(3\sqrt{3},\frac{\pi ^{c}}{4}\right)\)
• B. \(\left(3\sqrt{2},\frac{\pi ^{c}}{3}\right)\)
• C. \(\left(3\sqrt{2},\frac{\pi ^{c}}{4}\right)\)
• D. \(\left(2\sqrt{2},\frac{\pi ^{c}}{4}\right)\)

Hint:

Use r equal to the square root of x squared plus y squared and theta equal to arctan of y over x.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Polar coordinates \((r,\theta)\) are given by \(r=\sqrt{x^{2}+y^{2}}\) and \(\tan\theta=\dfrac yx\).

Step 2: Radial Distance
\(r=\sqrt{3^{2}+3^{2}}=\sqrt{18}=3\sqrt2\)

Step 3: Angle
\(\tan\theta=\dfrac33=1\).
The point is in the first quadrant, so \(\theta=\dfrac\pi4\).

Step 4: Compare the options
The polar coordinates are \(\left(3\sqrt2,\dfrac\pi4\right)\), option (C).

Final Answer:
The polar coordinates are 3 root 2 and pi over 4. \[ \boxed{\text{(C) }\left(3\sqrt{2},\dfrac{\pi ^{c}}{4}\right)} \]