Circle

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.

  • Center-Radius Form

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)}

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Posted in XI

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