Particular Cases of Circle

The locus of a point that moves in a plane so that its distance from a fixed point and the plane remains constant is called a circle. In other words, you can say a circle is a collection of all the points in a plane, each of which is at a constant distance from a fixed point in that plane. There is a fixed point in a circle, so the point of the circle is called the centre. 

The radius is the distance between the centre and the circle’s boundary. The radius of a circle cannot be negative; it is always positive. There are many different types of circles like concentric circles, congruent circles, circumscribed circles, cyclic quadrilateral, inscribed circles, and many more. 

Some important terms of the circle:

There are some important points of the circle which you have to understand carefully. However, they are:

• Interior of a circle
• The exterior of a circle
• Chord of a circle
• Diameter of a circle (length of diameter = 2 * radius) 
• The secant of a circle
• Tangent of a circle
• Arc of a circle
• Angle subtended by an arc
• The sector of a circle
• The segment of a circle (major segment and minor segment) 

Some particular cases of the circle:

Every particular case of a circle is important. Because every case contains different conditions and rules. However, let us discuss some special cases, which are as follows:-

The equation of a circle is (x-h)2 + (y-k)2 = r2. However, h and k are the circle’s centres, and r is the circle’s radius. So, if we have information about the coordinate and the centre of the circle, we can easily solve the equation. 

For e.g.:

If the centre of the circle is (1,2) and the circle’s radius is 4. So, find out the equation of the circle. 

As we know that the formula of the equation of a circle is (x-h)2 + (y-k)2 = r2

So, 

h= 1, k=2

r= 4

Now, putting the value in the formula of the equation of a circle, 

(x-1)2+(y-2)2 = 42

(x2−2x+1)+(y2−4y+4) =16

X2+y2−2x−4y-11 = 0 

Suppose the coordinate point of a circle is S(x,y). However, let us consider the circle’s radius, which is equal to ST. As we know, there are two coordinate points one is S(x,y) and the other one is the origin (0,0). These two coordinates will help us to find the distance between those two points. As a result, the equation of the circle from the centre to origin is:

                  x2+y2= a2

Some general forms of the equation of a circle

The general form of the equation of the circle is as follows:-

x2 + y2 + 2gx + 2fy + c = 0,  g, f, and c are some values.

On adding the values g2+ f2 on both sides of the equation. So, it will give the equation.

x2 + y2 + 2gx + 2fy + c+g2+f2-c= g^2 + f^2 − c ………………equation 1.

As (x+g)2 = x2+ 2gx + g2 and (y+f)2 =y2 + 2fy + f2

On substituting the values from equation (1)

Now we have, 

(x+g)2+ (y+f)2= g2 + f2−c …………….equation 2.

On comparing equation 2 with (x−h)2 + (y−k)2 = a2, 

where (h, k) is the circle’s centre and ‘a’ is the circle’s radius.

h=−g, k=−f

a2 = g2+ f2−c

g2 + f2 > c is the real radius of the circle. 

g2 + f2 = c, then the circle radius is zero. This shows us that the point of the circle coincides with the centre. However, these types of circles are known as point circles. 

g2 + f2 <c, so there is an imaginary circle radius. However, the circle contains an imaginary radius and a real centre. 

Some more particular cases of the circle:

The equation when the circle touch on the x-axis:-

                    (x±h)² + (y±k)² = k²

When the circle touches the y-axis, the question will be:-

                   (x±h)² + (y±k)² = h²

The condition when the circle touch on the x-axis from the origin the equation will be:-

             x² + (y±k)² = k²

             x² + y² + 2ky=0

The condition when the circle touch on the y-axis from the origin the equation will be:-

 (x ±h)² + y² = h²

 x² + y² + 2xh=0

Conclusion:

As there are many properties of a circle, there are standard forms of the equation of a circle, length of the tangent, the chord of the tangent, pair of tangents, intersection line of a circle, and many more. 

The circle is a topic that is used in physics as well. For example, you can find the uniform circular motion of a particle, uniform circular motion in an acceleration of a wave-particle, uniform motion of an artificial satellite, and many more.