Equations of a circle

The equation of a circle is an algebraic way to define a circle given the centre and radius length of the circle. This equation can be applied to a variety of circular 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 its centre, whereas the circumference of the circle is defined as the distance encircling the circle (c). The radius is the line that runs from the centre to one of the circumference’s points. 

Standard Equation of a Circle

The standard equation of a circle gives precise information on the circle’s centre and radius, making it much easier to understand the circle’s centre and radius at a glance. The usual equation for a circle with a radius of r and a centre at (a, b).

A circle can be depicted in a variety of ways, including:

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.

1. General form
2. Standard form
3. Parametric form
4. Polar form

General form

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 quickly convert from the general form to the standard form, we’ll use the completing the square formula.

Standard form

A circle is a group of points in a plane that are separated by a definite distance (the radius) from any point (the centre). The diameter is the length of a line segment that passes through the centre and ends at the circle’s endpoints. A circle can also be constructed by intersecting a cone with a plane that is perpendicular to the cone’s axis.

Parametric form

x=rcos , y=rsin is the parametric equation for the circle x2+y2=r2.

x=-g+rcos , y=-f+rsin are the parametric equations for the circle x2+y2+2gx+2fy+c=0.

Polar form

The equation for a circle with radius ‘ p’ and centred at the origin is x2+y2=p2. In the circle equation, substitute the values of x=rcos  and y=rsin .

Distance formula

This equals (x-0)2+(y-0)2=x2+y2 according to the distance formula. If and only if x2+y2=r, or if we square both sides:   x2+y2=r2 , a point (x,y) is at a distance r from the origin. This is the equation for a circle centred at the origin with radius r.

The Pythagorean Theorem was used to determine the lengths of the sides of a right triangle. This theorem will be used to find distances on a rectangular coordinate system. We can relate the geometry of a conic and algebra by determining distance on the rectangular coordinate system, which opens up a world of possibilities for application.

The first thing we’ll do is create a formula for calculating distances between points in a rectangular coordinate system. We’ll plot the points and draw a right triangle, much like we did in Graphs and Functions when we discovered slope. The Pythagorean Theorem is then used to determine the length of the hypotenuse of the triangle, which is the distance between the points.