All About General Form of the Equation of a Circle

In maths or geometry, a circle is a two-dimensional shape described as a locus of points equidistant from a fixed point known as the circle’s centre. A circle is just a round-shaped figure with no sides or edges in mathematics. This article will introduce you to the mathematical sections of a circle and its numerous varieties, as well as the various formulae for area, circumference, radius, centre, and more, along with properties and solved examples.

Definition of circle:-

In mathematics, the circle is a closed, two-dimensional curved figure. It can also be defined as a collection of points drawn at equal distances from the centre. The radius is the set distance between the centre and the circumference, and the diameter is the line that runs through the centre and connects two locations on the circumference.

Properties of Circles:-

The following are some of the most important properties of circles:

• The diameter of the circle divides it into two equal parts
• Congruent circles are those that have the same radius or diameter
• The circle’s diameter is twice its radius and is the longest chord
• Equal chords are always at the same distance from the circle’s centre
• A chord’s perpendicular bisector passes through the circle’s centre
• When two circles collide, the line that connects their intersecting points is perpendicular to the line that connects their centre points
• Circular shapes with varied measurements or radii/ diameters are comparable
• A perpendicular bisector of a circle’s chord is the radius
• The radius and the tangent always make a 90-degree angle
• If two tangents share a common origin, they are identical
• Equal circles have equal radii, as well as equal areas and circumferences
• The distance between the centre of a circle and the longest chord (diameter) is zero

Equation of Circle:-

The position of a circle on a cartesian plane is represented by a circle equation. Given the centre and radius of a circle, it can be drawn on a piece of paper. We can draw the circle on the cartesian plane using the equation of the circle once we determine the coordinates of the circle’s centre and radius. The equation of a circle can be expressed in a variety of ways as standard form, general form, polar form etc.

Standard form of equation of Circle:-

The standard equation of a circle provides exact information about the circle’s centre and radius, making it much easier to read the circle’s centre and radius at a glance. The usual equation for a circle with its centre at (h,k)and radius r is given by,

x-h2+y- k2= r2

 where (x, y) is any point on the circle.

General form of equation of circle:-

x² + y² + 2gx + 2fy + c = 0 is the general form of the circle equation. Where g, f, and c are constants, this general form is used to obtain the coordinates of the circle’s centre and radius. The general form of the equation of a circle, in contrast to the conventional form, makes it difficult to find any relevant properties about any given circle.

Derivation of general form from the standard form:-

As per the standard form of the equation of circle, the equation is as follows;

x-h2+y-k2= r2

Where, (x,y) represents the coordinate of any point on the circle.

              (h,k) represents the coordinates of centre

               r represents the radius of the circle

The above equation can also be written as;

x2+h2-2hx+ y2+ k2-2yk= r2

x2+ y2-2hx-2ky+h2+ k2– r2=0

Taking, -h = g and -k = f and h² + k² – r² = c, we get

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

Hence this is the required general form of the circle equation.

Coordinate of centre and radius from the general form:-

As per the general form of circle equation,

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

As we have already seen previously, we have assumed that

-h = g, which implies h = -g

-k = f, which implies k = -f

h² + k² – r² = c, which implies r=h2+ k2-c

Comparing this with the standard form, we get

The radius of circle is given as , r=h2+ k2-c

and the coordinate of the centre of the circle is (-g , -f) .

Conclusion:-

On a plane, a circle is a closed geometric object. A circle’s points are all the same distance apart from the circle’s centre, which is a fixed point. The circle’s centre is defined as the point from which all of the circle’s points are equidistant. The radius of a circle is the distance between its centre and any point on it. The circle’s radius is twice its diameter. A general equation, known as the equation of a circle, can be used to represent a circle in a cartesian coordinate system. In a polar and spherical coordinate system, a circle’s equation can also be generalised.