IGCSE Additional Mathematics: Coordinate Geometry of the Circle Practice Questions
A circle with centre (a, b) and radius r has equation (x minus a) squared plus (y minus b) squared equals r squared. The expanded form is found by completing the square in both variables.
Topic 9 of Cambridge IGCSE Additional Mathematics 0606 rests on moving between the two forms of the circle equation. Completing the square twice is the step every question depends on. The questions below cover both directions plus tangents.
What you need to know for Coordinate Geometry of the Circle
- Standard form(x minus a)2 + (y minus b)2 = r2, with centre (a, b) and radius r.
- Expanded formx2 + y2 + 2gx + 2fy + c = 0. Complete the square in x and in y to recover the centre and radius.
- Tangent and radiusA tangent is perpendicular to the radius at the point of contact, so their gradients multiply to minus 1.
- Line meeting a circleSubstitute the line into the circle equation. A positive discriminant gives two intersections, zero gives a tangent, negative gives none.
- Point on a circleA point lies on the circle if its coordinates satisfy the equation, or equivalently if its distance from the centre equals the radius.
IGCSE Additional Mathematics Coordinate Geometry of the Circle questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Find the centre and radius of the circle x2 + y2 minus 6x + 4y minus 12 = 0.
Show the worked answer
- Group the x terms and the y terms: (x2 minus 6x) + (y2 + 4y) = 12.
- Complete the square in x: x2 minus 6x = (x minus 3)2 minus 9.
- Complete the square in y: y2 + 4y = (y + 2)2 minus 4.
- Substituting gives (x minus 3)2 + (y + 2)2 = 12 + 9 + 4 = 25, so the centre is (3, minus 2) and the radius is 5.
Write down the equation of the circle with centre (2, minus 3) and radius 4, and expand it into the form x2 + y2 + 2gx + 2fy + c = 0.
Show the worked answer
- Standard form: (x minus 2)2 + (y + 3)2 = 42 = 16.
- Expand: x2 minus 4x + 4 + y2 + 6y + 9 = 16.
- Collect terms: x2 + y2 minus 4x + 6y + 13 = 16.
- Bring 16 across: x2 + y2 minus 4x + 6y minus 3 = 0.
A circle has centre (3, minus 2) and passes through the point P(7, 1). Find the equation of the tangent to the circle at P.
Show the worked answer
- Gradient of the radius from the centre to P: (1 minus (minus 2)) divided by (7 minus 3) = 3 divided by 4.
- The tangent is perpendicular to the radius, so its gradient is the negative reciprocal: minus four thirds.
- Use the point P(7, 1): y minus 1 = minus four thirds (x minus 7).
- Expand: y = minus four thirds x + 28 over 3 + 1.
- So y = minus four thirds x + 31 over 3. Rewriting as a check, 3y + 4x = 31, and substituting P gives 3 + 28 = 31.
Determine whether the line y = x + 6 intersects the circle x2 + y2 = 9.
Show the worked answer
- Substitute the line into the circle: x2 + (x + 6)2 = 9.
- Expand: x2 + x2 + 12x + 36 = 9, so 2x2 + 12x + 27 = 0.
- Calculate the discriminant: 122 minus 4 x 2 x 27 = 144 minus 216 = minus 72.
- The discriminant is negative, so the quadratic has no real solutions and the line does not meet the circle.
Common mistakes in this topic
- Giving r squared as the radius.
- Reversing the sign of the centre coordinates.
- Using the radius gradient for a tangent.
- Completing the square in only one variable.
- Answering an intersection question without calculating the discriminant.
Exam tips
- Complete the square in x and in y separately, then combine. Keep the two corrections on the right hand side.
- The centre coordinates are the opposite of the signs inside the brackets.
- Tangent gradient is the negative reciprocal of the radius gradient. Check the product equals minus 1.
- For intersection questions, always finish with the discriminant and a written conclusion.
- Verify a tangent equation by substituting the point of contact.
Practise 20 more questions like this, free
vStudyWise marks every answer instantly, tracks the topics you keep dropping marks on and turns them into a weekly study plan.
Coordinate Geometry of the Circle FAQs
How do I find the centre and radius from an expanded circle equation?
Group the x terms and y terms, complete the square in each, and move the correction constants to the right hand side. The equation then reads as a squared bracket in x plus a squared bracket in y equals r squared, from which the centre and radius are read directly.
How do I find the equation of a tangent to a circle?
Find the gradient of the radius joining the centre to the point of contact, then take its negative reciprocal for the tangent gradient. Use that gradient with the coordinates of the point of contact to write the equation of the line.
How do I tell whether a line meets a circle?
Substitute the equation of the line into the circle equation to obtain a quadratic in one variable, then calculate the discriminant. Positive means two points of intersection, zero means the line is a tangent, and negative means the line misses the circle entirely.
Why is the radius not the number on the right hand side?
The standard form has r squared on the right, not r. A right hand side of 25 therefore means a radius of 5. Forgetting to take the square root is one of the most frequent errors in this topic.
Continue through the IGCSE Additional Mathematics syllabus
Related IGCSE Additional Mathematics topics
See all 17 IGCSE Additional Mathematics practice topics ›
Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.