How to Find the Centre and Radius from the Equation of a Circle
The concept of Equation of a Circle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios—especially in coordinate geometry for classes 10, 11, and 12, as well as in entrance exams like JEE and NEET.
What Is the Equation of a Circle?
An Equation of a Circle is an algebraic way to express all the points that are a fixed distance (called the radius) from a single fixed point (called the centre). In coordinate geometry, this concept is used to solve problems about finding centres, radii, tangents, and intersections. You’ll find this concept used in topics such as coordinate geometry, conic sections, and in the study of tangents and normals.
Key Formula for Equation of a Circle
Here’s the standard formula for the equation of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where (h, k) is the centre of the circle and r is the radius.
The general form is:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
where centre = (−g, −f) and radius = \(\sqrt{g^2 + f^2 - c}\).
Cross-Disciplinary Usage
The equation of a circle is not only useful in Maths but also appears in Physics (circular motion, optics), Computer Science (graphics, game design), and even real-life architectural design. For students preparing for JEE or NEET, mastering the circle equation makes questions involving geometry and trigonometry much easier.
Step-by-Step Illustration
Let’s see an example: Find the centre and radius of the circle with equation \( x^2 + y^2 - 6x + 8y + 9 = 0 \).
1. Write the general form: \( x^2 + y^2 + 2gx + 2fy + c = 0 \)2. Compare coefficients: \( 2g = -6 \implies g = -3 \), \( 2f = 8 \implies f = 4 \), \( c = 9 \)
3. Centre is (−g, −f) = (3, −4)
4. Radius is \( \sqrt{(-3)^2 + (4)^2 - 9} = \sqrt{9 + 16 - 9} = \sqrt{16} = 4 \)
5. Final Answer: Centre = (3, –4), Radius = 4
Speed Trick or Vedic Shortcut
Here’s a handy speed trick for quickly converting from general form to standard form:
- Group x-terms and y-terms: \( x^2 - 6x + y^2 + 8y = -9 \)
- Complete the square for x and y:
\( x^2 - 6x + 9 + y^2 + 8y + 16 = -9 + 9 + 16 \)
\( (x - 3)^2 + (y + 4)^2 = 16 \) - Now centre = (3, -4), radius = 4 (as in the previous example)
With practice, you can do this mentally to save crucial seconds during exams! Vedantu classes share more such quick tips and live practice for exam speed and clarity.
Try These Yourself
- What is the equation of a circle with centre (2, –1) and radius 5?
- Find the centre and radius of \( x^2 + y^2 + 4x - 10y + 13 = 0 \).
- Convert \( (x + 2)^2 + (y - 7)^2 = 16 \) to general form.
- Write the equation of a circle that passes through (0,0), (4,0), and (0,4).
Frequent Errors and Misunderstandings
- Forgetting to reverse the sign for centre when comparing general form coefficients.
- Mixing radius and diameter or misunderstanding formula for radius in general form.
- Missing negative signs when squaring terms while converting from general to standard form.
Relation to Other Concepts
The idea of equation of a circle connects closely with topics such as equation of a line and conic sections. Mastering this helps you understand tangents, lengths of chords, and even circles in 3D (sphere equations) in later chapters.
Classroom Tip
An easy way to remember the standard circle equation: “A circle is all (x, y) points that are exactly a distance r from the centre (h, k).” Use graph sheets to plot one yourself. Vedantu’s interactive online classes often use circle-drawing tools to make this concept visual and memorable.
We explored the Equation of a Circle—from definition, formula, worked examples, speed mistakes, and connections to other maths branches. Keep practising with Vedantu’s resources and circle equation worksheets to get confident and exam-ready!
Explore more related topics for deeper understanding:
FAQs on Equation of a Circle – Formula, Forms, and Examples
1. What is the standard form of the equation of a circle as per the Class 11 syllabus?
The standard form of a circle's equation is (x - h)² + (y - k)² = r². In this equation, the point (h, k) represents the coordinates of the circle's centre, and r represents the length of its radius. This form is the most direct way to describe a circle in a coordinate plane.
2. How is the general form of a circle's equation, x² + y² + 2gx + 2fy + c = 0, different from the standard form?
The key difference lies in how they present information. The standard form, (x - h)² + (y - k)² = r², explicitly reveals the circle's centre (h, k) and radius r. In contrast, the general form, x² + y² + 2gx + 2fy + c = 0, does not. To find the centre and radius from the general form, you must first convert it to the standard form by completing the square.
3. How do you find the coordinates of the centre and the length of the radius from the general equation of a circle?
For a circle given by the general equation x² + y² + 2gx + 2fy + c = 0, you can find the centre and radius directly using these formulas:
- The centre of the circle is at the coordinates (-g, -f).
- The radius of the circle is calculated as r = √(g² + f² - c).
These are derived by comparing the general equation to the expanded standard form.
4. What is the step-by-step process to find the equation of a circle when the endpoints of its diameter are given?
If you are given the endpoints of a diameter, say (x₁, y₁) and (x₂, y₂), follow these steps:
- Step 1: Find the Centre. The centre of the circle (h, k) is the midpoint of the diameter. Use the midpoint formula: h = (x₁ + x₂)/2 and k = (y₁ + y₂)/2.
- Step 2: Find the Radius. The radius (r) is the distance from the centre to either of the endpoints. Use the distance formula to calculate it.
- Step 3: Write the Equation. Substitute the values of h, k, and r into the standard form equation: (x - h)² + (y - k)² = r².
5. How do you derive the equation of a circle that passes through three given points?
To find the equation of a circle passing through three non-collinear points, you start with the general form, x² + y² + 2gx + 2fy + c = 0. Substitute the (x, y) coordinates of each of the three points into this equation. This will create a system of three linear equations with three variables (g, f, and c). Solving this system will give you the values for g, f, and c, which you can then place back into the general equation to define the circle.
6. What is a common mistake students make when identifying the centre from the general equation of a circle?
A very common mistake is forgetting to reverse the signs of the coefficients. The centre's coordinates are (-g, -f), not (g, f). For an equation like x² + y² - 8x + 10y - 12 = 0, we have 2g = -8 and 2f = 10. This means g = -4 and f = 5. Therefore, the centre is at (-g, -f), which is (4, -5), not (-4, 5).
7. How is the distance formula fundamental to deriving the standard equation of a circle?
The standard equation of a circle is a direct application of the distance formula. A circle is defined as the set of all points (x, y) that are at a constant distance (the radius, r) from a fixed point (the centre, (h, k)). The distance formula calculates the distance between two points. By setting the distance between (x, y) and (h, k) equal to r, we get r = √[(x - h)² + (y - k)²]. Squaring both sides of this equation gives us the standard form: (x - h)² + (y - k)² = r².
8. For the general equation x² + y² + 2gx + 2fy + c = 0 to represent a circle, what is the condition for the radius to be a real, positive value?
The radius of a circle in the general form is calculated as r = √(g² + f² - c). For this to represent a real circle, the radius 'r' must be a positive real number. This is only possible if the expression inside the square root is positive. Therefore, the condition is that g² + f² - c > 0. If g² + f² - c = 0, the equation represents a single point (a point-circle), and if it is less than 0, it represents an imaginary circle with no real points.
9. What are some real-world applications of the equation of a circle in fields like physics and computer graphics?
The equation of a circle is used extensively in many fields:
- Physics: It is used to describe the circular path of planets, electrons orbiting a nucleus, and the propagation of circular waves from a point source.
- Computer Graphics: Programmers use the equation to render circular elements in user interfaces, create collision-detection zones in games, and design animations.
- Engineering: It is fundamental in designing gears, tunnels, pipes, and other cylindrical or spherical components.
10. Why is mastering the different forms of a circle's equation crucial for solving problems in conic sections and for competitive exams like JEE?
Mastering the circle's equation is crucial because the circle is the foundational concept within the broader topic of conic sections. A strong understanding of its forms, properties, and relationship with tangents and lines is essential for tackling more complex conics like parabolas, ellipses, and hyperbolas. For competitive exams like JEE, questions often integrate concepts, requiring you to use the circle equation to solve problems involving tangents, chords, and intersections, making it a high-weightage and indispensable topic.
