Circle is the locus of a point in a plane, such that it is equidistant from a fixed point. The fixed point is known as the center of the circle and the distance is known as the radius of the circle.
If the center of the circle is and radius is , then the equation of circle is given by
This is the distance formula, applied to the condition of the locus of a point.
Let and be endpoints of any diameter of a circle. Then, the diameter form is given by
This is based on the fact that a semicircular arc subtends an angle of at the center and by inscribed angle theorem, the angle subtended is .
General Equation of Circle in 2 Dimensions
The general equation of a circle is given by
Its center is and radius is .
Key Points :
I) The equation does not contain any term containing .
II) Coefficients of the terms and are same. If they aren’t, the equation does not represent a circle.
III) If , the equation does not represent a circle.
Parametric Equations of a Circle
The equations containing and are known as equations in Cartesian form. We’ve another form of equations, known as parametric equations, where both and are represented in terms of a parameter . Thus, when the parameter changes its value, and $y$ change accordingly. If the circle has radius $a$ and center at origin, then the parametric equations are given by
If the center is at , then