Understanding the Centre and Radius of a Circle

In coordinate geometry, a circle is defined as the set of all points in a plane that are equidistant from a fixed point. This fixed point is called the centre of the circle, and the constant distance is known as the radius. Understanding these two fundamental properties is crucial for analyzing and manipulating circle equations.

The Standard Equation of a Circle

The most common form of a circle's equation in coordinate geometry is the standard form. If a circle has its centre at the point $(h, k)$ and a radius of $r$ , its equation is given by:

$(x - h)^2 + (y - k)^2 = r^2$

Here, $(x, y)$ represents any point on the circumference of the circle.

What are the two key parameters that define a circle in coordinate geometry?

The centre and the radius.

Identifying the Centre and Radius

From the standard equation $(x - h)^2 + (y - k)^2 = r^2$ , we can directly identify the centre and radius:

The centre is $(h, k)$ and the radius is $r$.

In the equation $(x - h)^2 + (y - k)^2 = r^2$ , the values of $h$ and $k$ directly give the coordinates of the centre, and the square root of the constant term on the right side gives the radius.

To find the centre $(h, k)$ , observe the terms being subtracted from $x$ and $y$ . If the equation is in the form $(x + a)^2 + (y + b)^2 = c$ , then $h = -a$ and $k = -b$ . The radius $r$ is found by taking the square root of the constant term on the right side of the equation. It's important to remember that the equation is $r^2$ , so you must take the square root to find $r$ . The radius must always be a positive value.

Remember: If the equation is given as $(x+h)^2$ or $(y+k)^2$ , the corresponding coordinate of the centre is $-h$ or $-k$ , respectively.

Example: Finding Centre and Radius

Consider the equation of a circle: $(x - 3)^2 + (y + 5)^2 = 16$ .

Comparing this to the standard form $(x - h)^2 + (y - k)^2 = r^2$ :

We see that $h = 3$ and $k = -5$ . Therefore, the centre of the circle is $(3, -5)$ .

Also, $r^2 = 16$ . Taking the square root of both sides, we get $r = \sqrt{16} = 4$ . So, the radius of the circle is 4.

The standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$ , visually represents the Pythagorean theorem. For any point $(x, y)$ on the circle, the horizontal distance from the centre $(h, k)$ is $|x - h|$ and the vertical distance is $|y - k|$ . These distances form the legs of a right-angled triangle, with the radius $r$ as the hypotenuse. The equation is a direct application of $a^2 + b^2 = c^2$ , where $a = |x - h|$ , $b = |y - k|$ , and $c = r$ . This geometric interpretation reinforces why the equation is structured this way and how the centre and radius are embedded within it.

📚

Text-based content

Library pages focus on text content

General Form of a Circle's Equation

Sometimes, a circle's equation is given in the general form: $x^2 + y^2 + 2gx + 2fy + c = 0$

To find the centre and radius from this form, we need to convert it back to the standard form by completing the square.

Rearranging the terms: $(x^2 + 2gx) + (y^2 + 2fy) = -c$ .

Completing the square for $x$ : $(x^2 + 2gx + g^2) = (x + g)^2$ . We add $g^2$ to both sides.

Completing the square for $y$ : $(y^2 + 2fy + f^2) = (y + f)^2$ . We add $f^2$ to both sides.

The equation becomes: $(x + g)^2 + (y + f)^2 = g^2 + f^2 - c$ .

Comparing this to the standard form $(x - h)^2 + (y - k)^2 = r^2$ :

General Form Parameter	Standard Form Equivalent	Interpretation
$2g$	$-2h$	Coefficient of $x$
$2f$	$-2k$	Coefficient of $y$
$c$	$-r^2 + h^2 + k^2$	Constant term

Therefore, from the general form $x^2 + y^2 + 2gx + 2fy + c = 0$ :

Centre is $(-g, -f)$ .

Radius is $r = \sqrt{g^2 + f^2 - c}$ .

Important: For the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ to represent a real circle, the condition $g^2 + f^2 - c > 0$ must be satisfied. If $g^2 + f^2 - c = 0$ , it represents a point circle, and if $g^2 + f^2 - c < 0$ , it represents an imaginary circle.

What is the condition for

x^2 + y^2 + 2gx + 2fy + c = 0

to represent a real circle?

$g^2 + f^2 - c > 0$

Learning Resources

NCERT Mathematics Class 11 - Chapter 10: Straight Lines(documentation)

This official NCERT textbook chapter covers coordinate geometry, including detailed explanations and examples of circles, their equations, centre, and radius.

Khan Academy: Circle Equation(video)

A clear video tutorial explaining the standard equation of a circle and how to identify its centre and radius.

Byju's: Circle - Equation, Properties, Examples(blog)

This article provides a comprehensive overview of circles in coordinate geometry, including the standard and general forms of their equations and how to find the centre and radius.

Vedantu: Equation of a Circle(blog)

Learn about the different forms of the equation of a circle and how to derive the centre and radius from them with solved examples.

Maths is Fun: Equation of a Circle(documentation)

An easy-to-understand explanation of the circle equation, including interactive elements and clear definitions of centre and radius.

Toppr: Centre and Radius of a Circle(blog)

This resource offers a practical example of finding the centre and radius of a circle from its general equation, demonstrating the process of completing the square.

JEE Main Mathematics: Coordinate Geometry - Circles(video)

A YouTube video specifically tailored for JEE Main preparation, covering circles and their properties, including centre and radius identification.

Wikipedia: Circle(wikipedia)

The Wikipedia page for 'Circle' provides a broad mathematical context, including its definition, properties, and various equations in different coordinate systems.

Brilliant.org: Circle Equation(documentation)

Brilliant.org offers an interactive approach to learning the equation of a circle, focusing on the geometric intuition behind the formula.

StudyIQ IAS Academy: Coordinate Geometry - Circles(video)

This video provides a detailed explanation of circles in coordinate geometry, suitable for competitive exam preparation, with a focus on centre and radius.