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3 years ago in Computational Geometry , Mathematics Education By Srajan

How can one prove that the equation of a circle is y = mx + c?

While tutoring undergraduates, I encountered a persistent misconception: the belief that a circle can be expressed by the linear equation y = mx + c. I want to address this definitively. Could you outline the correct proof for a circle's equation to help me clearly demonstrate why the linear form is fundamentally incompatible?

CoordinateGeometry CircleEquation

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By Shobha Answered 2 years ago

I've seen this confusion often. The key is to return to the geometric definition. A circle is the set of all points equidistant (the radius r) from a fixed point (the center (h,k)). The proof applies the Pythagorean distance formula: the horizontal and vertical legs of the right triangle formed by a point (x,y) and the center must satisfy (x-h)² + (y-k)² = r². A linear equation like y=mx+c describes a constant slope, which cannot encapsulate the constant distance property of a circle. They are fundamentally different objects.

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