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Proofs : Circles and Irrational Numbers

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I. A circle has center (2, 4).
1. Prove that and are not both on the circle.
2. Prove that if the radius is less than 5, then the circle does
not intersect the line given by .
(Please give formal proofs for part 1 & 2.)

II. Prove that is irrational.
(Please prove this by contradiction.)

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Proofs involving circles and irrational numbers are provided. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Problem #1
1. Proof: (by contradiction)
The circle is centered at . If both and are on this circle, then and they are equal to the radius of the circle. From the distance formula, we have

So . We get a contradiction.
Therefore, ...

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