Family Of Circles

When a group of circles has the same properties, they are referred to as a family of circles. It belongs to the field of coordinate geometry. We will study the formulas for the family of circles and the family circle of notes and solve instances in this chapter. x2 + y2 + 2gx + 2fy + c = 0 is the generic equation for a circle. We will also discuss methods for locating the family of circles under specific circumstances.

Let S = x2 + y2 + 2gx + 2fy + c = 0 be the solution.

S1 = x2 + y2 + 2g1x + 2f1y + c1 = 0

S2 = x2 + y2 + 2g2x + 2f2y + c2 = 0

L = lx + my + n = 0

A Family of Circles with a Constant Centre

The formula is written as (x-h)2 + (y-k)2 = r2

(h, k) are constants, but r is a variable. This is the formula for a concentric family of circles. Fixing the radius will result in a concentric circle.

A Family of Circles passes via the Junction of a Line and a Circle

S+λL = 0, where λ is a parameter, is the needed equation for the family of circles travelling through the point of intersection of circle S = 0 and line L = 0.

At a Certain Point, a Family of Circles Touches a Line

At a point (x1, y1) on any finite m, the equation of a family of circles meeting the line y-y1 = m(x-x1) is given by

+ (x-x1)2 (y-y1) [(y-y1) – m(x-x1)] + 2 equals 0

The equation becomes (x – x1)2 + (y-y1)2 + (x-x1) = 0 if m is infinite.

A Family of Circles that Passes Across Two Spots

The equation for the family of circles travelling through two specified points, P(x1, y1) and Q(x2, y2) 

A Family of Circles that Passes via the Point of Intersection of Two Circles

The family of circles going through the point of intersection of two circles S=0 and S ′ =0 has the equation

S+S = 0 is a formula that can be used to calculate the sum of two numbers

= -1, where is a parameter.

The formula of a circle bounding a quadrilateral with lines expressing the sides in order L1 = 0, L2 = 0, L3 = 0 and L4 = 0.

The required family of circles equation is as follows: L1L3 + λL2L4 = 0, which provides a coefficient of x2 = coefficient of y2 and coefficient of XY = 0.

Circles Types

Concentric Circles: Circles with the same centre are known as concentric circles. An annulus is an area between two concentric circles with different radii. Any two circles can be made concentric by inverting any two circles and selecting the inversion centre as one of the limiting points. Consider a circle with a radius of (g2+f2-c) and a centre of (-g, -f). The circle’s equation is as follows:

x2+y2+2gx+2fy+c = 0

x2+y2+2gx+2fy+c1 = 0 is the equation of a concentric circle to the above circle.

The centre of both circles will be the same (-g, -f), but the radius will be different. (c) (c1) 

The Annulus has a ring-like flat form. Finding the areas of the outer and inner circles can be used to compute the annulus’ area. To achieve the answer, we must find the difference between the areas of both circles.

Contact Circles: Contact of circles occurs when the outer surfaces of two circles come into contact.

Case 1: When the total of two circles’ radii equals the distance between their centres, the circle’s contact is on the outside. The circles will meet the criterion.

c1c2 = r1 + r2

Case 2: The circles will contact internally if the radii differ by the same amount as the distance between the centres, and the circles will meet the criterion.

c1c2 = r1 – r2

Orthogonal Circles: A circle that is orthogonal to another circle has a 90° angle between them. Circles are considered to be orthogonal when this criterion is met. As a result, they are also known as perpendicular circles.

Properties of Circles

• If the radii of two circles are the same, they are said to be congruent.

• A circle’s diameter is the length of its most extended chord.

• Same chords subtend equal angles at the centre of a circle.

• The radius bisects chords drawn perpendicular to it

• The radii of circles with varying radii are similar.

Lambda in the Family of Circles

The equation for the family of circles going through the point of intersection of two circles S=0 and S′=0 is as follows. S+λS′=0. where is a parameter, and is equal to 1

When the Circle Crosses the Y-axis

Once the circle crosses the y-axis, h = a. If a circle meets the y-axis, the circle’s radius is equal to the x-coordinate of the centre.

Conclusion

A family of circles refers to a group of circles that share the same features. It falls within the category of coordinate geometry, and a family of circles is a long list of circles.