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5.Circle – Standard Equations
Class 11 • Mathematics • NCERT
Circle – Standard Equations
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This lesson introduces the concept of a circle and its standard equations as per NCERT Class 11. Students learn how to identify the centre and radius and convert equations into standard form confidently.
Lesson Objectives
- Understand the definition of a circle.
- Learn the standard equation of a circle.
- Identify centre and radius from an equation.
- Convert general form into standard form.
1. Definition of a Circle
A circle is the set of all points in a plane which are at a constant distance from a fixed point.
• Fixed point → Centre
• Constant distance → Radius
• Constant distance → Radius
2. Standard Equation of a Circle
If the centre of a circle is at (h, k) and radius is r, then its equation is:
(x − h)² + (y − k)² = r²
3. Circle with Centre at Origin
If the centre of the circle is at the origin (0,0), then the equation becomes:
x² + y² = r²
4. General Form of a Circle
The general form of the equation of a circle is:
x² + y² + 2gx + 2fy + c = 0
Centre = (−g, −f)
Radius = √(g² + f² − c)
5. Simple NCERT Examples
• Equation of circle with centre (2, −3) and radius 5:
(x − 2)² + (y + 3)² = 25
(x − 2)² + (y + 3)² = 25
• For x² + y² − 4x + 6y − 3 = 0
Centre = (2, −3)
Practice Questions (NCERT)
- Define a circle.
- Write the standard equation of a circle.
- Write the equation of a circle with centre at origin and radius 7.
- What is the centre of x² + y² + 2x − 4y + 1 = 0?
- Write the general form of a circle.
- Find the radius of x² + y² − 6x + 8y − 11 = 0.
- What does r represent in circle equation?
- Can radius be negative?
- How many parameters define a circle?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- Set of all points at a constant distance from a fixed point.
- (x − h)² + (y − k)² = r²
- x² + y² = 49
- (−1, 2)
- x² + y² + 2gx + 2fy + c = 0
- √(9 + 16 + 11) = 6
- Radius of the circle.
- No
- Three (h, k, r)
- Yes
© Aviate Learning – Conic Sections (NCERT Class 11)