# Circle

• ### Prerequisites : Locus and Line

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 ${(h,k)}$ and radius is ${r}$, then the equation of circle is given by

${(x-h)^2 + (y-k)^2 = r^2}$

This is the distance formula, applied to the condition of the locus of a point.

• ### Diameter Form

Let ${(x_1,y_1)}$ and ${(x_2, y_2)}$ be endpoints of any diameter of a circle. Then, the diameter form is given by

${(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0}$

This is based on the fact that a semicircular arc subtends an angle of ${\pi^c}$ at the center and by inscribed angle theorem, the angle subtended is ${\frac {\pi}{2}}$.

• ### General Equation of Circle in 2 Dimensions

The general equation of a circle is given by

${x^2 + y^2 + 2gx + 2fy + c =0}$

Its center is ${(-g,-f)}$ and radius is ${\sqrt {g^2 + f^2 -c}}$.

• ### Key Points :

I) The equation does not contain any term containing ${xy}$.

II) Coefficients of the terms ${x^2}$ and ${y^2}$ are same. If they aren’t, the equation does not represent a circle.

III) If ${g^2 + f^2 -c < 0}$, the equation does not represent a circle.

• ### Parametric Equations of a Circle

The equations containing ${x}$ and ${y}$ are known as equations in Cartesian form. We’ve another form of equations, known as parametric equations, where both ${x}$ and ${y}$ are represented in terms of a parameter ${\theta}$. Thus, when the parameter changes its value, ${x}$ and $y$ change accordingly. If the circle has radius $a$ and center at origin, then the parametric equations are given by

${x = a \ cos (\theta) \ and \ y = a \ sin (\theta)}$

If the center is at ${(h,k)}$, then

${x = h + a \ cos (\theta) \ and \ y = k + a \ sin (\theta)}$