Circles

Important Formulas:

Standard Forms of a Circle:

























12. For circle , parametric equation is 

General Equation of a circle:

13. The general equation of a circle is given by  whose centre of the circle 

14.  Radius of the circle 

15. If   , then the radius  of the circle is real and hence the circle is also real.

16. If  , then the radius of  the circle is  and the circle is known as point circle.

17. If , then the radius of the circle is imaginary. Such a circle is imaginary, which is not possible to draw.

Position of a Point with Respect to a Circle:

18. A point  lies outside on   or  inside a circle according as where 

19. Intercepts on the Axes: The length of the intercepts made by the circle    with   and  - axis are  and 

20. If   then the roots of the equation   are real  and distinct, so the  circle   meets the  in two real and distinct points.

21. If  , then the roots of the equation    are real and equal, so the circle touches axis, then intercept on axis is 

22. If , then the roots of the equation    are imaginary, so the given circle does meet axis in real point. Similarly, the circle  cuts the axisin real and distinct points touches or does not meet in real point according 

Equation of tangent: A line which touch only one point of a circle.

23. Point form  the equation of the tangent at the point  to a circle  is 

24. Point form the equation of the tangent at the point   to a circle  is .

25. Slope  form The equation of the tangent of slope   to the circle  are 

26. Slope form the equation of the tangents  of slope  to the circle  are    and the coordinates of the points of contact are

27. Slope form The equation of the tangents of slope  to the circle  are  and the coordinates  of the point of contact are 

28. Parametric from the equation of the tangent to the circle  at the point   is .

Equation of normal: A line which is perpendicular to the tangent is known as a normal.

29. Point form : The equation of normal at the point  to the circle  is   or 

30. Point form : The equation of normal at the  point  to the circle   is 

31. Slope form: The equation of a normal of slope   to the circle  is .

32. Para metric form: the equation of normal to the circle  at the point  is   or 

33. The line  meets the circle in unique real point  ot touch the circle  , if 

and the point of contacts are 

34. The line  touches the circle  , if .

35. The point of intersection of the tangent at the points  and   on the circle   is given by

    and   

36. Power of a point   with respect to the circle  is .

Pair of tangents:

37. The combined equation of the pair of tangents drawn from a point  to the circle  is

  or  

where,   and 

38. The length of the tangents from the point  to the circle   is equal to 

39. Chord of contact   of two tangents, drawn from   to the circle        is   or  Similarly, for the circle   is 

40. Equation of chord Bisected at a Given point The equation of chord of the circle  bisected at the point  is given by  i.e 

41. Director Circle The locus of the point of intersection of two perpendicular tangents  to a given circle is called a director circle. For circle  , the equation of director of director circle is .

42. Common Chord The chord joining the points  of intersection of two given circles is called common chord. If      and   be two circle, such that   and , then their common chord is given by  

43. Common Chord The chord joining the points  of intersection of two given circles is called common chord. If   denote the centre of the given circles, then their common chord  

44. Common Chord The chord joining the points  of intersection of two given circles is called common chord. If   and   be the radii of  two circles, then length of common chord is 

45. Angle of intersection of two circles 

for_CIRCLES