CIRCLE

In this chapter we are going to learn the definition of a 'circle' and discuss its general equation, its tangents and normals and various other properties like the power of a point, pole  and polar with respect to a circle. Relative positions of two Circles and their common tangents.

The locus of a point in a plane which moves in such a way that its distance from a fixed point in the plane is always a constant is called a circle.

• The equation of a circle with its center at C(a, b ) and radius r is:        (x – a)² + (y – b)² = r²
• If a= b = 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).
• The parametric equations of a circle with centre (h, k) and radius 'r' are given by x=h+rcos, y=k+rsin
• 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.
• Equal chords of a circle subtend equal angles at the centre.
• The angle subtended by an arc at the centre id double the angle subtended by the same arc at the circumference of the circle.
• The angle between the tangent and the radius is 90°.
• Angles in the same segment are equal.
• The angle in a semi-circle is 90°.
• Two angles at the circumference subtended by the same arc are equal.
  • The equation of the tangent on a point on the circle and the equations of the chord of contact are both represented by T = 0. But the difference between the two is that in case of chord of contact, the point say (x₁, y₁) lies outside the circle while in case of tangent it lies on the circle.
  • Common tangents may be either direct or transverse.  
  • Direct common tangents to two circles meet on the line joining the centres and hence divide it externally in the ratio of their radii. 
  • Transverse common tangents also meet on the line of centres as the direct common tangents but the difference being that they divide it internally in the ratio of their radii.

•                                                    

  • There cannot be any common tangent if a circle lies completely inside another without touching it.
  • If two circles touch each other internally one and only one common tangent is possible.
  • If two circles touch each other in two distinct points, then in such a case, two common tangents can be drawn.
  • In the case of two circles touching each other externally, three common tangents can be drawn.
  • Four common tangents are possible when two circles neither touch nor intersect but one lies completely inside the other.
  • How to calculate direct common tangents:

  • Length of C₁C₂ > |r₁ – r₂|
  • For two circles with equations S and S’, where

  S ≡ x² + y² + 2gx + 2fy + c and

  S’ ≡ x² + y² + 2g’x + 2fy’ + c’

  Then equation of radical axis of two circles S = 0 and S’ = 0 is given by S = S’

  2(g – g’)x + 2(f – f’)y + (c – c’) = 0

  • The condition for two circles with centres at C₁(x₁, y₁) and C₂(x₂, y₂) and radii r₁, r₂ respectively, to touch each other internally and externally a

  1. Internally: If |C₁ C₂| = |r₂ – r₁| and the point of contact is ((r₁x₂-r₂x₁)/(r₁+r₂) , (r₁y₂-r₂y₁)/(r₁+r₂)).
  2. Externally: If |C₁ C₂| = |r₂ + r₁| and the point of contact is ((r₁x₂+r₂x₁)/(r₁+r₂) , (r₁y₂+r₂y₁)/(r₁+r₂)).

  • For the circle S = 0, the equation, S – S’ = 0 represents the equation of the radical axis i.e. the equation of radical axis is

  2x(g – g’) + 2y(f – f’) + c – c’ = 0

  • If the circles S = 0 and S’ = 0 intersect in real and distinct points, then S – S’ = 0 is the equation of the common chord of the two circles.

  If S’ = 0 and S = 0 touch each other, then S – S’ = 0 is the equation of the common tangent to the two circles at the point of contact.

  

  • The radical axis of two circles is perpendicular to the line joining the centres.

  1. 1. The radical axis of three circles taken two at a time are concurrent and the point of concurrency is known as the radical centre.
     2. The radical axis of two circles bisects their direct common tangents.
     3. If two circles cut the third circle orthogonally, then the radical axis of the two circles will pass through the centre of the third circle.
     4. If in a system of circles, every two of them have the same radical axis, then the system is a coaxial system

The equations of the family of circles in various cases:

• 1. 1. S₁ + λ S₂ = 0 is the family of circles passing through the intersection of S₁ = 0 and S₂ = 0.
     2. The family of circles passing through the intersection of line L and circle S is given by S + λL = 0.  A particular value of λ in the equation yields a unique circle.
     3. Family of circles passing through two given points A(x₁, y₁) and B(x₂, y₂) are given by the equation

                                                 

  • Two circles S = 0 and S₁ = 0 with centres at (–g, –f) and (–g₁, –f₁) cut each other orthogonally if the following equation is satisfied:

  2gg₁+ 2ff₁ = c + c

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  Relative position of two circles:

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  1). If One circle lies  completely inside the other circle

  

  Number of common tangents = 0.

   

  2). If two circles touch each other  internally

  .

  A number of common tangents =1.

   

  3). If two circles touch each other externally

  .

  A number of common tangents = 3.

   

  4). If two circles intersect at two points

  

  Number of common tangents = 2.

   

  5). If two circles do not touch each other

  .

  The number of common tangents = 4.

RADII AND CHORD

We begin by recapitulating the definition of a circle and the terminology used for circles. Throughout this module, all geometry is assumed to be within a fixed plane.

• A circle is the set of all points in the plane that are a fixed distance (the radius) from a fixed point (the centre).
• Any interval joining a point on the circle to the centre is called a radius. By the definition of a circle, any two radii have the same length. Notice that the word ‘radius’ is being used to refer both to these intervals and to the common length of these intervals.
• An interval joining two points on the circle is called a chord.
• A chord that passes through the centre is called a diameter. Since a diameter consists of two radii joined at their endpoints, every diameter has a length equal to twice the radius.
• The word ‘diameter’ is used to refer both to these intervals and to their common length.
• A line that cuts a circle at two distinct points is called a secant. Thus a chord is an interval that the circle cuts off a secant, and diameter is the interval cut off by a secant passing through the centre of a 'circle'.