Mastering the General Equation of a Circle

Welcome to the fascinating world of coordinate geometry! In this module, we'll delve into the general equation of a circle, a fundamental concept for competitive exams like JEE. Understanding this equation allows us to identify and analyze circles based on their properties.

From Standard to General Form

Recall the standard equation of a circle with center $(h, k)$ and radius $r$ : $(x-h)^2 + (y-k)^2 = r^2$ . Expanding this equation gives us a more general form.

The general equation of a circle is a quadratic equation in two variables, x and y.

Expanding $(x-h)^2 + (y-k)^2 = r^2$ yields $x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2$ . Rearranging terms, we get $x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0$ .

Let's expand the standard form: $(x-h)^2 + (y-k)^2 = r^2$ . This becomes $x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2$ . To get the general form, we move all terms to one side: $x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0$ . This equation is typically written as $x^2 + y^2 + 2gx + 2fy + c = 0$ , where $g = -h$ , $f = -k$ , and $c = h^2 + k^2 - r^2$ .

The General Equation: $x^2 + y^2 + 2gx + 2fy + c = 0$

The general equation of a circle is given by: $x^2 + y^2 + 2gx + 2fy + c = 0$ . This form is incredibly useful because it directly reveals the circle's center and radius.

The general equation of a circle, $x^2 + y^2 + 2gx + 2fy + c = 0$ , can be transformed back into the standard form $(x-h)^2 + (y-k)^2 = r^2$ by completing the square. By rearranging and adding $g^2$ and $f^2$ to both sides, we get $(x^2 + 2gx + g^2) + (y^2 + 2fy + f^2) = g^2 + f^2 - c$ . This simplifies to $(x+g)^2 + (y+f)^2 = g^2 + f^2 - c$ . Comparing this to the standard form, we can identify the center as $(-g, -f)$ and the radius as $\sqrt{g^2 + f^2 - c}$ . For a valid circle, the term under the square root, $g^2 + f^2 - c$ , must be positive.

📚

Text-based content

Library pages focus on text content

Key Properties from the General Equation

Property	General Equation: $x^2 + y^2 + 2gx + 2fy + c = 0$
Center	( $-g$ , $-f$ )
Radius	$\sqrt{g^2 + f^2 - c}$

Remember: 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 hold. If $g^2 + f^2 - c = 0$ , it represents a point circle (a single point), and if $g^2 + f^2 - c < 0$ , it represents an imaginary circle (no real points).

Identifying Circle Properties: An Example

Let's consider the equation $x^2 + y^2 - 6x + 8y - 11 = 0$ . To find its center and radius, we compare it to the general form $x^2 + y^2 + 2gx + 2fy + c = 0$ .

In the equation

x^2 + y^2 - 6x + 8y - 11 = 0

, what are the values of

g

f

, and

c

$2g = -6 \implies g = -3$ , $2f = 8 \implies f = 4$ , and $c = -11$ .

Now, we can find the center and radius:

Center = $(-g, -f) = (-(-3), -4) = (3, -4)$ .

Radius = $\sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + (4)^2 - (-11)} = \sqrt{9 + 16 + 11} = \sqrt{36} = 6$ .

Special Cases and Conditions

The general equation $Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0$ (where $A \neq 0$ ) also represents a circle. To use the formulas for center and radius, we first divide the entire equation by $A$ to get $x^2 + y^2 + \frac{2G}{A}x + \frac{2F}{A}y + \frac{C}{A} = 0$ . Here, $g = G/A$ , $f = F/A$ , and $c = C/A$ . The conditions for a real circle, point circle, and imaginary circle remain the same with these adjusted values.

Crucially, for an equation to represent a circle, the coefficients of $x^2$ and $y^2$ must be equal and positive, and there should be no $xy$ term.

Key takeaway: The general equation $x^2 + y^2 + 2gx + 2fy + c = 0$ is a powerful tool for quickly determining a circle's center and radius, essential for solving many coordinate geometry problems in competitive exams.

Learning Resources

General Equation of a Circle - Concepts and Formulas(documentation)

Provides a clear explanation of the general equation of a circle, its derivation, and formulas for center and radius.

Circle: General Equation - Maths for JEE(blog)

Explains the general equation of a circle with examples and highlights key properties and conditions.

Coordinate Geometry: The Circle - Khan Academy(video)

An introductory video on circles in coordinate geometry, covering standard and general forms.

General Equation of a Circle - Toppr(blog)

Details the general equation of a circle, including how to find the center and radius and conditions for a circle.

Circle - General Equation - Doubtnut(video)

A video explanation focusing on the general equation of a circle and its applications.

JEE Mathematics: Circle - General Equation(documentation)

While this link is for straight lines, it's part of a comprehensive NCERT solution set that often includes circles and their equations in context for competitive exams.

Circle (Coordinate Geometry) - Wikipedia(wikipedia)

Provides a broad overview of circles, including their mathematical properties and equations in various contexts.

Coordinate Geometry - Circles - StudyIQ IAS(video)

A video tutorial covering circles in coordinate geometry, likely including the general equation for exam preparation.

JEE Main 2024 Maths: Circle - Important Topics & Formulas(blog)

Highlights important topics and formulas for circles in JEE Main, including the general equation.

Understanding the General Equation of a Circle - Math Stack Exchange(paper)

A forum discussion that often delves into the nuances and derivations of mathematical concepts like the general equation of a circle.