Equation of Circle

The equation of a circle in simple words is a technique to represent a circle in an algebraic form and for this equation, there is a requirement of two parameters that are center and radius of the circle. Circles use different formulae for determining their area and circumference than they do for calculating the area and circumference of a circle. This equation is used in a number of circle issues in coordinate geometry, and it is very useful. 

In order to represent a circle in the Cartesian plane, the equation of the circle must be used. If we know the center and radius of a circle, we can draw it on a piece of paper. Similarly, we can draw a circle on a Cartesian plane if we know the coordinates of the center and radius. 

What is a circle?

A circle is a shape formed by all points in a plane that are at a set distance from a given point, the center; alternatively, it is the curve sketched out by a point that travels in a plane while maintaining a constant distance from a given point, the circumference. The radius of a circle is defined as the distance between any point on the circle and its center. On most occasions, it is necessary for the radius to be a positive value. 

A circle is an example of a degenerate situation. Unless otherwise stated, this article is about circles in Euclidean geometry, and in particular, the Euclidean plane, unless otherwise specified. 

Specific to the plane of the plane, a circle is a simple closed curve that separates it into two distinct regions: the interior and the outside. If a circle is simply the border of a figure, the word “circle” may be used interchangeably to refer to either that boundary or to the whole figure including its interior; nevertheless, in precise technical terminology, the circle is only the boundary and the entire form is known as a disc. 

Another way to define a circle is as a special kind of ellipse in which the two foci are coincident, the eccentricity is zero, and the semi-major and semi-minor axes are equal; or as the two-dimensional shape that contains the greatest amount of area per unit perimeter squared, as determined by the calculus of variations. 

how to use the equation of a circle?

The position of a circle in the Cartesian plane is represented by a circle equation. We can write the equation of a circle if we know the coordinates of the circle’s center and the length of its radius. All of the points on the circumference of the circle are represented by the circle equation.

The locus of points whose distance from a fixed point is constant is represented by a circle. The constant value is the radius r of the circle, and this fixed point is termed the center of the circle. The standard equation of a circle with centre at (x₁,y₁) and radius r is given as (x-x₁)²+(y-y₁)²=r²

Equation of a circle formula

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 different forms.

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

Let’s take a closer look at the two most common forms of the equation of circle, the general and standard forms, as well as the polar and parametric forms.

General Equation of Circle

The general form of the equation of circle is written as x²+y²+2gx+2fy+c=0 where g, f, and c are constants, this general form is used to obtain the coordinates of the circle’s centre and radius. The generalised form of the equation of a circle, in contrast to the standard form,

makes it difficult to find any relevant properties about any given circle. So, to quickly convert from the general form to the standard form, we’ll use the completing the square formula.

Standard 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 standard equation of circle with centre at (x₁,y₁) and radius r is (x-x₁)²+(y-y₁)²=r² where (x, y) is an arbitrary point on the circumference of the circle. 

The distance between this point and the centre of the circle is equal to the radius of the circle, as shown in the diagram. Let us use the distance formula to calculate the distance between these two places. 

√(x-x₁)²+(y-y₁)²=r

Squaring both sides, we get the standard form of the equation of the circle as:

(x-x₁)²+(y-y₁)²=r²

Parametric Equation of Circle

We know that the general form of the equation of a circle is x²+y²+2hx+2ky+c=0. We start with a broad point on the circle’s periphery, such as, say, the centre (x, y). The line connecting this approximate location with the centre of the circle (-h, -k) m. akes an angle of θ. The parametric equation of a circle can be written as x²+y²+2hx+2ky+c=0 where x=-h+rcosθ and y=-k+rsinθ.

Polar Equation of Circle

The polar form of the circle equation is nearly identical to the parametric form of the circle equation. For a circle centred at the origin, we commonly write the polar form of the equation of the circle. Take a point P(rcosθ, rsinθ) on the circle’s periphery, where r is the point’s distance from the origin. We know that the equation of a circle with radius ‘p’ and centred at the origin is x²+y²=p

Substitute the value of x = rcosθ and y = rsinθ in the equation of circle.

(rcosθ)²+(rsinθ)²=p²

r²cos²θ+rsin²θ=p²

r²(cos²θ+sin²θ)=p²

r²(1)=p²

r²=p²