The Pythagorean Theorem And The Equation Of A Circle In Cartesian Coordinates.

Equation of a Circle

A way to express the definition of a circle on the coordinate plane.

The equation of a circle in Cartesian coordinates (x, y) is (x – a)^2 + (y – b)^2 = r^2, where (a, b) is the center of the circle and r is the radius.

To derive this equation, we start with the Pythagorean theorem, which says that for any point on the circle, the distance to the center of the circle is equal to the radius:

d^2 = (x – a)^2 + (y – b)^2

where d is the distance between the point (x, y) and the center of the circle (a, b).

Squaring both sides and simplifying, we get:

(x – a)^2 + (y – b)^2 = r^2

This is the equation of a circle with center (a, b) and radius r. The equation can also be expressed in the form x^2 + y^2 + 2gx + 2fy + c = 0, where the center of the circle is given by (-g, -f) and the radius is given by r=sqrt(g^2+f^2-c). This form of the equation is helpful when solving problems involving circles in homogeneous coordinates.

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