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?
The standard form of the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) represents the coordinates of the circle's center, and r represents its radius. This form is straightforward and clearly shows the circle's center and radius.
2. How do I find the radius and center from the circle’s equation?
To find the radius and center from a circle's equation, follow these steps:
- If the equation is in standard form [(x - h)² + (y - k)² = r²], the center is (h, k), and the radius is r.
- If the equation is in general form [x² + y² + 2gx + 2fy + c = 0], then complete the square to convert it to standard form. The center will be (-g, -f), and the radius is √(g² + f² - c).
3. What is the general form of a circle's equation in Maths?
The general form of a circle's equation is x² + y² + 2gx + 2fy + c = 0, where g, f, and c are constants. This form doesn't directly reveal the center or radius, but it can be converted into the standard form using the method of completing the square.
4. How to convert the general form to standard form?
To convert the general form (x² + y² + 2gx + 2fy + c = 0) to standard form [(x - h)² + (y - k)² = r²], complete the square for both x and y terms. Rearrange the equation to group x terms and y terms together. Add and subtract (g)² to the x terms and (f)² to the y terms to create perfect squares. The resulting equation will be in the standard form, revealing the center (-g, -f) and radius √(g² + f² - c).
5. Is the equation of a circle important for JEE and CBSE?
Yes, understanding the equation of a circle is crucial for both JEE and CBSE examinations. It forms a fundamental part of coordinate geometry, and related questions frequently appear in various exam formats (MCQs, long-answer type).
6. How do I find the equation of a circle given its center and radius?
If you know the center (h, k) and radius r of a circle, you can directly write its equation in standard form: (x - h)² + (y - k)² = r².
7. How do I find the equation of a circle given three points on the circle?
Given three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) on a circle, you can find its equation by substituting the coordinates into the general form (x² + y² + 2gx + 2fy + c = 0). This will result in a system of three equations with three unknowns (g, f, and c). Solve this system to obtain the values of g, f, and c. Then, rewrite the equation in standard form to find the center and radius.
8. What is the equation of a circle passing through the origin?
A circle passing through the origin will have its equation in the general form with c = 0. The general form simplifies to x² + y² + 2gx + 2fy = 0. This equation can be converted to standard form to identify the center and radius.
9. What is the equation of a circle with a diameter given by two points?
If you are given two endpoints (x₁, y₁) and (x₂, y₂) of a circle's diameter, first find the midpoint of the diameter using the midpoint formula which is the circle's center. Then, find the distance between the midpoint and one of the endpoints, which represents the radius. Finally, use the standard form (x-h)²+(y-k)² = r² to determine the equation, where (h,k) is the center and r is the radius.
10. Can the equation of a circle have complex numbers for its center or radius?
No, in standard coordinate geometry, a circle's center and radius must be real numbers. Complex numbers are not typically used in representing circles in this context.
11. How does the equation of a circle change when it's tangent to a line?
When a circle is tangent to a line, the distance from the circle's center to the line is equal to the circle's radius. This condition can be used to find the equation of the circle, involving the line's equation and the circle's center and radius. The point of tangency lies on both the circle and the line.
12. What are some real-world applications of the equation of a circle?
The equation of a circle has numerous real-world applications, including:
- Engineering: Designing circular components, analyzing circular motion.
- Physics: Describing orbits of planets and satellites, analyzing circular waves.
- Computer Graphics: Drawing circles and other circular shapes.
- Mapping and Navigation: Locating positions using coordinates and distances.
