Polar Equation: Convergence

Algebra Level 4

What is the maximum value of n n that causes the graph of the polar equation below to converge for 0 θ 10 π ? 0 \leq \theta \leq 10\pi?

Note: A graph converges if it can be bounded by a circle with finite perimeter for a given domain.

2 -2 1 -1 π 4 \pi-4 0 0 1 1 π \pi

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1 solution

Timothy Cao
Apr 1, 2018

[Amateur's explanation]

Consider the Cartesian version of this polar equation: (easier to type)

y = ( x 4 ) 1 ( x 4 ) + n y = (x-4)^{\sqrt{\frac{1}{(x-4)+n}}}

This is an equivalent function

y = e l n ( x 4 ) ( 1 ( x 4 ) + n ) y = e^{ln(x-4) (\frac{1}{(x-4)+n})}

We can simply find when the exponent that is l n ( x 4 ) ( 1 ( x 4 ) + n ) ln(x-4) (\frac{1}{(x-4)+n}) diverges to positive infinity

(If it diverges to negative infinity, the entire power goes to 0)

Considering the bounds 0 θ 10 π 0 \leq \theta \leq 10\pi ,

The above expression only diverges to positive infinity while ( x 4 ) < 1 (x-4) < -1

Note: We ignore the restriction from the ln because we created that restriction

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