Circles

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The equation of a circle with its center at C(x0, y0) and radius r is:

        (x – x0)2 + (y – y0)2 = r2

  • If x0 = y0 = 0 (i.e. the centre of the circle is at origin) then equation of the circle reduce to x2 + y2 = r2.
  • If r = 0 then the circle represents a point or a point circle.
  • The equation x2 + y2 + 2gx + 2fy + c = 0 is the general equation of a circle with centre (–g, –f) and radius √(g2+f2-c).
  • Equation of the circle with points P(x1, y1) and Q(x2, y2) as extremities of a diameter is (x – x1) (x – x2) + (y – y1)(y – y2) = 0.
  • For general circle, the equation of the chord is x1x + y1y + g(x1 + x) + f(y1 +y) + c = 0
  • For circle x2 + y2 = a2, the equation of the chord is x1x + y1y = a2
  • The equation of the chord AB
  • (A ≡ (R cos α, R sin α); B ≡ (R cos β, R sin β)) of the circle x2 + y2 = R2 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(x1, y1) lies outside, on, or inside a circle  ?

           S ≡ x2 + y2 + 2gx + 2fy + c = 0, according as S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c > = or < 0.

  • The equation of the chord of the circle x2 + y2 + 2gx + 2fy +c=0 with M(x1, y1) as the midpoint of the chord is given by:  

           xx1 + yy1 + g(x + x1) + f(y + y1) = x12 + y12 + 2gx1 + 2fy1  i.e. T = S1 

  • 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