Centre and radius of a circle: Class 11 practice
x² + y² + 2gx + 2fy + c = 0 is a circle with centre (−g, −f) and radius √(g² + f² − c). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Completing the square
The idea behind centre and radius of a circle
x² + y² + 2gx + 2fy + c = 0 is a circle with centre (−g, −f) and radius √(g² + f² − c). Completing the square gives the same result.
Step-by-step method
- Make the x² and y² coefficients 1.
- Read off g and f (halve the x and y coefficients).
- Centre (−g, −f), radius √(g² + f² − c).
Worked example 1 (Basic)
Find the centre and radius of x² + y² + 12x − 6y + 9 = 0.
- g = 6, f = -3, c = 9
- Centre (-6, 3)
- r = √(36 + 9 − (9)) = 6
- Answer: Centre (-6, 3), radius 6
Worked example 2 (Standard)
Find the centre and radius of x² + y² − 8x + 0y − 65 = 0.
- g = -4, f = 0, c = -65
- Centre (4, 0)
- r = √(16 + 0 − (-65)) = 9
- Answer: Centre (4, 0), radius 9
Worked example 3 (Challenge)
Find the equation of the circle with centre (-2, -5) passing through (4, -5).
- r = 6
- (x + 2)² + (y + 5)² = 36
- Answer: (x + 2)² + (y + 5)² = 36
Common mistakes and misconceptions
- Sign of the centre coordinates
- Forgetting to subtract c
Practice questions
- (Basic) Find the centre and radius of x² + y² + 4x − 2y − 11 = 0.
- (Basic) Find the centre and radius of x² + y² + 8x − 12y + 27 = 0.
- (Standard) Find the centre and radius of x² + y² − 4x − 6y + 12 = 0.
- (Standard) Find the centre and radius of x² + y² − 4x − 10y + 4 = 0.
- (Challenge) Find the equation of the circle with centre (6, -4) passing through (15, -4).
- (Challenge) Find the equation of the circle with centre (-2, 0) passing through (-1, 0).
Answer key with full working
- Q1: g = 2, f = -1, c = -11 → Centre (-2, 1) → r = √(4 + 1 − (-11)) = 4 → Answer: Centre (-2, 1), radius 4
- Q2: g = 4, f = -6, c = 27 → Centre (-4, 6) → r = √(16 + 36 − (27)) = 5 → Answer: Centre (-4, 6), radius 5
- Q3: g = -2, f = -3, c = 12 → Centre (2, 3) → r = √(4 + 9 − (12)) = 1 → Answer: Centre (2, 3), radius 1
- Q4: g = -2, f = -5, c = 4 → Centre (2, 5) → r = √(4 + 25 − (4)) = 5 → Answer: Centre (2, 5), radius 5
- Q5: r = 9 → (x − 6)² + (y + 4)² = 81 → Answer: (x − 6)² + (y + 4)² = 81
- Q6: r = 1 → (x + 2)² + (y + 0)² = 1 → Answer: (x + 2)² + (y + 0)² = 1
Syllabus note
This skill is listed under "Conic Sections" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is centre and radius of a circle in?
"Conic Sections", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.