Circles

Introduction:

Using the equation of a circle, a person can express the shape of a circle in terms of its centre and radius. A circle is the set of all points in a plane situated at a fixed distance from a fixed point in the plane. The fixed point in the process is called the centre of the circle. Distance from the centre of the circle to any point on the circle is called the circle’s radius. All equations related to circles represent the points located at the circumference or the circle’s boundary.

Cartesian equations in a circle can be algebraically determined by quickly drawing a circle using its centre and radius on the cartesian plane. There are different forms of circles like general, standard, parametric, and polar form.

Body : 

Equation of a Circle

On the Cartesian plane, a circle must be represented by its equation. Drawing a circle is simple if we know its centre and radius. We can draw a circle on a Cartesian plane if we know the centre and radius coordinates. One can portray a circle in several ways, such as:

• General form
• Standard form
• Parametric form
• Polar form

General equation

For a circle, general equation is represented as x2 + y2 + 2ax + 2by + c = 0

Where, 

x = point at the boundary

y = point at the boundary

a = constant

b = constant

c = constant

Centre of this circle = (-a, -b)

Radius of the circle = √(a2 + b2 – c)

Standard equation

The standard equation of a circle is represented as:

(x – a)2 + (y – b)2 = r 2

Where, 

r = radius of the circle

(a,b) = centre of the circle 

This equation is also termed as circle-radius form. 

(x, y) are the point at the boundary or circumference of the circle

Parametric equation 

The parametric equation of a circle can be written as x² + y² + 2hx + 2ky + C = O where x = -h+ rcos θ and y = -k + rsinθ. 

Equation of a circle with centre on X-axis

Consider (x,y) as arbitrary points on the circle’s circumference. 

The circle‘s centre is on the x-axis (-a,0) with radius r.

The equation is

x2 + y2 + 2ax + c = 0

Equation of a circle with centre on Y-axis

(x,y) are arbitrary points on the circumference of the circle. The circle’s centre is on the y-axis (0, -b) with radius r.

1. The general equation is:

 x2 + y2 + 2by + c = 0

1. The circle passing through the origin,The equation becomes

x2 + y2 + 2ax + 2by = 0 and c=0

1. The equation for a circle passing through the origin that cuts the x-axis at (a,0) and the y-axis at (0,b) is:

 x2 + y2 – ax -by = 0

1. The circle touches the Y-axis at origin and is centred at X-axis.

 The equation is x2 + y2 – 2ax = 0

 ( 0,0) and (2a, 0) are the endpoints of the diameter of the circle

1. The circle touching X-Axis at origin and centred at Y-Axis

 The equation is x2 + y – 2by = 0

 (0,0) and (0, 2b) are the endpoints of the diameter

Essential Notes on Equation of Circle

Here is a list of a few points that should be remembered while studying the equation of a circle

• A circle’s equation always begins with x2 + y2 in its general form.
• If a circle crosses both the axes, then there are four places of intersection between the circle and the axes.
•  Only two points of contact exist if a circle contacts both axes.
• x2+y2+axy+C=0 is not the circle’s equation; since No xy term can be found in the circle equation.
•  R and e are always used to express the equation of the circle when it is presented in this form.
• The radius of a circle is the distance from the centre to any point on its perimeter. Thus, the radius of the circle is always a positive number. 

Conclusion :

The circle radius is the distance from the circle’s centre to the boundary, so its values always remain positive. There are different cartesian equations of a circle depending upon the circle’s coordinates. It helps to find the centre and radius of the circle in the cartesian plane. One form of the equation can be turned into another form with the help of the formulae using the values given in the questions.