Standard Form of Equation of A Circle

Given the circle’s centre and radius, the equation of the circle provides an algebraic technique to describe the circle. The equation of circle can be found in various circles issues in coordinate geometry. The equation of the circle is essential to know because it helps us to represent the circle on the Cartesian plane. If we know the centre 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 centre and radius coordinates. The equation of any circle can be represented in a variety of ways as follows:

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

Definition:-

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 centre and the length of its radius. The circle equation represents all of the points on the circumference of the circle.

A circle indicates the locus of points with a constant distance from a fixed point. The constant value is the radius r of the circle, and this fixed point is termed the centre of the circle. The standard circle equation with the centre at (h , k) and radius r is given as (x – h)² + (y – k)² = r² .

General equation of a circle:-

X2 + y2 + 2gx + 2fy + c = 0 is the generic 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 generic 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. So, to convert the general equation  of the circle to the standard form, we will use the method of completing the square.

Standard equation of a 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 classic circle equation with the centre at (h , k) and radius r is (x – h)² + (y – k)² = r², where (x, y) is a random point on the circle.

The radius of the circle is equal to the distance between this point and the centre. Let’s use the distance formula to calculate the distance between these sites.

i.e,  √r=(x-h)²+(y- k)²

It can also be depicted as follows;

      r²=(x-h)²+(y- k)²

Which is the required standard equation of the circle.

Parametric equation of circle:-

We know that x² + y² + 2hx + 2ky + C = 0 is the general version of the circle equation. Let’s imagine we start at a general position on the circle’s edge (x, y). The angle formed by the line connecting this general point and the circle’s centre (-h, -k) equals θ .Then circle’s parametric equation is 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 similar to that of  the parametric form of the circle. For a circle centred at the origin, we commonly write the polar version of the equation of the circle. Take a point on the circle’s boundary, P(rcosθ, rsinθ), where r is the point’s distance from the origin. The equation for a circle with radius ‘p’ and centred at the origin is x² + y² = p².

Putting the value of x = rcosθ and y = rsinθ in the above equation of circle, we get,

(rcosθ)² + (rsinθ)² = p²

r² cos²θ + r² sin²θ = p²

r² (cos²θ + sin²θ) = p²

r² (1) = p²

r = p , where p is assigned as the radius of the circle.

Formula for the radius of a circle:-

As we have already read about the standard equation of a circle, we can derive the formula for the radius of any circle from that formula.

According to the standard equation of circle, 

                                  r²=x-h²+y- k²

 where, r is the radius of the given circle.

              (h,k) = coordinate of the centre of the circle

              (x,y) = coordinate of any arbitrary point on the circle

Now taking the square root of both side of this equation we get,

                                      r= √(x-h)²+(y- k)²

Hence, this is the required formula of the radius of the circle.

Conclusion:-

An algebraic way to define a circle is presented, given the circle’s centre and radius length. This equation is applied to numerous circle problems in coordinate geometry. The diameter of the circle is defined as a line that connects two points on the circle’s perimeter and passes through the centre, while the circumference of the circle is defined as the distance encircling the circle . The radius is the line that runs from the centre to one point on the circumference.