**The equation of a **circle with its center at C(x_{0}, y_{0}) and radius r is:

(x – x_{0})^{2} + (y – y_{0})^{2} = r^{2}

- If x
_{0}= y_{0}= 0 (i.e. the centre of the circle is at origin) then**equation of the circle**^{2}+ y^{2}= r^{2}**.** - If r = 0 then the circle represents a point or a point circle.
- The equation x
^{2}+ y^{2}+ 2gx + 2fy + c = 0 is the general equation of a circle with centre (–g, –f) and radius √(g^{2}+f^{2}-c). - Equation of the circle with points P(x
_{1}, y_{1}) and Q(x_{2}, y_{2}) as extremities of a diameter is (x – x_{1}) (x – x_{2}) + (y – y_{1})(y – y_{2}) = 0. - For general circle, the equation of the chord is x
_{1}x + y_{1}y + g(x_{1}+ x) + f(y_{1}+y) + c = 0 - For circle x
^{2}+ y^{2}= a^{2}, the equation of the chord is x_{1}x + y_{1}y = a^{2} - The equation of the
**chord**AB - (A ≡ (R cos α, R sin α); B ≡ (R cos β, R sin β)) of the circle x
^{2}+ y^{2}= R^{2}is given by x cos ((α + β )/2) + y sin ((α – β )/2) = a cos ((α – β )/2) - Chords are equidistant from the center of a circle if and only if they are equal in length.

Equal chords of a circle subtend equal angles at the center

- The angle subtended by an arc at the center id double the angle subtended by the same arc at the circumference of the circle.
- Angle between the tangent and the radius is 90°.
- Angles in the same segment are equal.
- Angle in a semi-circle is 90°.
- Two angles at the circumference subtended by the same arc are equal.

The below table describes the equations of circle according to changes in radii and centers:

## Formula Tables

- The point P(x
_{1}, y_{1}) lies outside, on, or inside a circle**?**

** **S ≡ x^{2} + y^{2} + 2gx + 2fy + c = 0, according as S_{1} ≡ x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c > = or < 0**.**

- The equation of the
**chord of the circle**x^{2}+ y^{2}+ 2gx + 2fy +c=0 with M(x_{1}, y_{1}) as the midpoint of the**chord**is given by:

** xx _{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) = x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} i.e. T = S_{1} **

- In case the radius and the central angle of a triangle are given, the length of the chord can be computed using the formula

Length of the chord = 2r sin (c/2), where ‘c’ is the central angle and ‘r’ is the radius

if there is a circle which has one tangent and one secant, then the square of the tangent is equal to the product of the secant segment and its external segment.

- If a radius or the diameter of a circle is perpendicular to a chord, then it divides the chord into two equal parts. The converse also holds true.

Hence, in the below figure, if OB is perpendicular to PQ, then then *PA* = *AQ.*

chord passes through the center of a circle. In the figure given below, OA is the perpendicular bisector of chord PQ and it passes through the circle. Similarly, OB is the perpendicular