The Standard Form of Equation of a Circle

A circle is defined as the locus of a point moving in a plane, such that its distance from a fixed point in the plane (i.e., the center) is constant for all such points. The equation of a circle is the algebraic way of expressing a circle if its center and length of the radius are given.

To represent the position of a circle in a Cartesian plane, an equation of a circle is required. The standard form of the equation of a circle is heavily used in coordinate geometry. This equation is different from the formulas used to calculate the circumference or area. 

What is a Circle and Equation of a Circle?

  • A circle is defined as the locus of a point moving in a plane, such that its distance from a fixed point in the plane (i.e., the center) is constant for all such points.
  • It is a closed, two-dimensional shape formed by tracing a point moving in a plane such that its distance from a fixed point is constant. The fixed point is the ‘centre of a circle, and the length of any line drawn from the circle’s center to its boundary is called the ‘radius of a circle.
  • The length of the longest line drawn in a circle is the diameter, i.e., a line segment having endpoints as the circle’s boundary while passing through the center.
  • The equation of a circle is the algebraic way of expressing a circle if its centre and length of the radius are given.
  • To represent the position of a circle in a Cartesian plane, an equation of a circle is required. The equation of a circle is heavily used in coordinate geometry.
  • The circle equation is different from the formulas used to calculate the circumference or area of a circle.

Different Forms of the Equation of a Circle

  • The equation of a circle is the algebraic way of representing the circle’s position in a Cartesian plane.
  • A circle is drawn on paper as long as its centre and the length of the radius is known.
  • Once the coordinates of the circle’s centre and the length of its radius are found using the equation of a circle, one can draw a circle on the Cartesian plane.
  • There are several ways to represent the equation of a circle. They are:
  • Standard form
  • General Form
  • Parametric Form
  • Polar Form
  • The two most common ways of representing a circle are – the general form of the equation of a circle and the standard form of the equation of a circle.
  • The general equation of a circle is: x² + y² + 2gx + 2fy + c = 0, where g, f, and c are constants.

Standard Forms of a Circle

There are various standard forms to represent a circle in a plane. Considering (x, y) as an arbitrary point on the circumference of a circle, the center of the circle as (h, k), and the length of the radius as r in a Cartesian plane, the standard forms of a circle are:

  • Equation of a circle with centre (h, k) is: (x – h)² + (y – k)² = r². If the centre of the circle is its origin i.e., centre is (0, 0), the equation of the circle is x² + y² = r².
  • If the circle passes through the origin, then the equation is: x² + y² – 2hx – 2ky = 0. 
  • The equation of a circle touching the X-axis in a plane is: x² + y² — 2hx — 2ry + h² = 0.
  • The equation of a circle touching the Y-axis in a plane is: x² + y² — 2rx — 2ky + k² = 0.
  • The equation of a circle touching both the X and Y-axes in a plane is: x² + y² — 2rx — 2ry + r² = 0.
  • The equation of a circle passing through the origin and the center lying on the X-axis is: x² + y² – 2rx = 0
  • The equation of a circle passing through the origin and the centre lying on the Y-axis is: x² + y² – 2ry = 0
  • The equation of a circle passing through the origin and cutting intercepts a and b on the coordinate axes is: x² + y² — by = 0.
  • The equation of a circle when the coordinates of endpoints of diameter are (x1, y1) and (x2, y2) is: (x — x1) (x — x2) + (y – y1) (y — y2) = 0.

Standard form Equation of a Circle

  • The standard form of equation of a circle gives precise information about the circle’s centre and its radius, making it easier to read the centre and radius at a glance.
  • Considering (x, y) as an arbitrary point on the circumference of a circle, the center of the circle as (h, k), and the length of the radius as r in a Cartesian plane, the standard form of the equation of a circle is (x – h)² + (y – k)² = r².
  • The distance between this arbitrary point P(x, y) and the center C(h, k) is equal to the radius r of the circle. By applying the distance formula between these points, we get | CP | = r, i.e.,

√{ (x – h)² + (y − k)² }=r

  • On squaring both the sides, we get the standard form of the equation of a circle, i.e.

(x – h)² + (y – k)² = r².

Conclusion

This article covers the concepts, formulas, and scientific terms devised to provide a standard form of the equation of a circle. We introduced the concepts about what a circle is and the equation of a circle to provide the foundation for the central topic, that is, the standard form of the equation of a circle. You are also made familiar with the different forms of equations of a circle and the standard form of a circle, along with formulas for better and easy understanding. In the FAQ section, some solved problems are provided for reference.

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