How do you derive the equation of a circle $(x−a)^2+(y−b)^2=r^2$ if a point on the y-axis is chosen as then you cannot form a triangle and as a result not apply Pythagoras' theorem and derive the equation of a circle.
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What do you mean with "a point on the y-axis is chosen"? – Raskolnikov Apr 14 '15 at 21:58
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The distance formula coming from the Pythogrean theorem is true even if one or both legs of the "triangle" with corners $(a, b)$ and $(x, y)$ have length zero. – Andrew D. Hwang Apr 14 '15 at 21:59
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If you consider a point with the same x-coordinate as the centre of the circle (x= a) then the equation just gives $(y-b)^2 = r^2$ ie y = b +/- r. It's still a solution to Pythagorus, just Pythagorus applied to a "triangle" with one side equal to zero so the hypotenuse is equal to the other side. – IanF1 Apr 14 '15 at 22:02