Types of Circles and their Significance

A circle, in particular, is a closed curve that splits the plane into two regions: interior and exterior. In ordinary usage, the phrase “circle” can refer to either the figure’s boundary or the entire figure, including its inside; in strict technical terms, the circle is merely the boundary, and the entire figure is referred to as a disc.

A circle is also a special type of ellipse in which the two foci are coincident, the eccentricity is 0, and the semi-major and semi-minor axes are equal; or the two-dimensional shape enclosing the maximum area per unit perimeter squared, using calculus of variations.

Types of circles

1. Concentric circles: Circles with the same centre are known as concentric circles. An annulus is the area between two concentric circles with different radii. By picking the inversion centre as one of the limiting points, inversion can be utilised to make any two circles concentric.
2. Contact of circles: If two circles with centres C₁x₁y₁ and C₂(x₂y₂) and radii r₁ and r₂ respectively, touch  each other externally, C₁C₂=r₁+r₂. Coordinates of the point of contact are A=

The circles touch each other internally if, C₁C₂=r₁–r₂. Coordinates of the point of contact are T=

3. Orthogonal circles: When two circles are sliced orthogonally, they form orthogonal curves and are referred to as orthogonal circles. A circle that is orthogonal to another circle has a 90-degree angle between them. The circles are considered to be orthogonal when this criterion is met. As a result, they are also known as perpendicular circles. Under inversion, orthogonal circles are invariant. Problem of Plane Strain in Plasticity is one of the many applications of orthogonal circles. S₁=0 and S₂=0 is given by S₁+λS₂=0, Where λ≠-1. 

Circle family

1. Family of circles having a fixed centre

The equation is given by x-h²+y-k²=r²

Here (h,k) is fixed and r is a varying parameter. This is the equation of the family of concentric circles.

2. Family of circles passing through the point of intersection of line and circle

The equation for the circle S=0 family going through the point of intersection and line L=0 is given by S+λL=0, 

Where is a parameter.

3. Family of circles passing through the point of intersection of two circle

S₁+λS₂=0, Where λ≠-1, is the equation of the family of circles passing through the point of intersection of two circles S₁=0 and S₂=0

4. Family of circles passing through two given points

The equation of the family of circles passing through two given points P(x₁,y₁) and Q(x₂,y₂) is given by 

(x-x₁)(x-x₂)+(y-y₁)(y-y₂)+ λx y1 x₁y₂ 1 x₂ y₂ 1 =0

5. Family of circles touching a line at a given point  

• The family of circles touching the line y-y₁=mx-x₁ at a point (x₁,y₁)

on any finite m has the equation(x-x₁)²+(y-y₁)²+λy-y₁-m(x-x₁)=0

If m is infinite, the equation becomes (x-x₁)2+(y-y₁)²+λx-x₁=0.

Chord of two circles 

Chord: The chord of the circle is a line segment that connects two places on the circle’s circumference. One of the line segments that can be drawn in a circle that has its endpoints on the circumference is the chord. To find the chord PQ, look at the circle below. A chord that goes through the centre of the circle is also known as a diameter.

Properties of the chord of a circle: A few key properties of a circle’s chords are listed below.

• The chord is bisected by a perpendicular drawn from the centre of the circle.
• The chords of a circle that are equidistant from the circle’s centre are equal.
• There is only one circle that travels through three intersecting points.
• When a circle chord is drawn, it divides the circle into two sections, which are referred to as the major and minor segments of the circle.
• A secant is formed when a chord is prolonged indefinitely on both sides.

Common Chord of two circles: We can connect the two crossing sites to form a common chord when two circles intersect.

The connecting line will be a perpendicular bisector of the common chord if we link the centres of these two circles.

At locations A and B, the circles O and Q intersect. Demonstrate that the line OQ connecting the two circles’ centres bisects AB and is perpendicular to it.

Conclusion

A circle is a planar shape formed by a closed curve with each point on the curve being the same distance from the circle’s centre. Everyone uses the concept of circles in geometry on a daily basis. It’s employed in agriculture, education, and architecture.