Polar Coordinates

Geometry Level 3

Find the polar coordinates, ( r , θ ) (r,\theta) of the centre of the circle r = cos ⁡ θ + sin ⁡ θ r = \cos\theta + \sin\theta .

( 1 2 , 0 ) \left( \frac1{\sqrt2} , 0 \right) ( 1 2 , π 2 ) \left( \frac12 , \frac \pi2 \right) ( 1 2 , π 4 ) \left( \frac1{\sqrt2} , \frac\pi 4 \right) ( 1 , π ) (1,\pi )

Sourasish Karmakar by Sourasish Karmakar , 19, India

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1 solution

Multiply thorough by r r to get

r 2 = r cos ⁡ ( θ ) + r sin ⁡ ( θ ) ⟹ x 2 + y 2 = x + y ⟹ ( x − 1 2 ) 2 + ( y − 1 2 ) 2 = 1 2 r^{2} = r\cos(\theta) + r\sin(\theta) \Longrightarrow x^{2} + y^{2} = x + y \Longrightarrow (x - \frac{1}{2})^{2} + (y - \frac{1}{2})^{2} = \frac{1}{2} .

This then describes a circle with radius 1 2 \frac{1}{\sqrt{2}} and center ( 1 2 , 1 2 ) (\frac{1}{2}, \frac{1}{2}) , which in polar coordinates is ( 1 2 , π 4 ) \boxed{(\frac{1}{\sqrt{2}}, \frac{\pi}{4})} .

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