Skills covered: Circle equations; completing the square; diameter; point location; equations with no real circle; line–circle intersections; tangent equations; circle geometry.
Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.
These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if circle still feels shaky; if you are confident, go straight to Section C and time yourself.
Section A
Prerequisite core skills
Revise earlier learning
Question 1
[2 marks]
Find the equation of the circle with centre (−2,3) and radius 5.
Question 2
[4 marks]
Find the centre and radius of x2+y2−6x+4y−12=0.
Question 3
[4 marks]
A(1,−2) and B(7,6) are the endpoints of a diameter. Find the equation of the circle.
Question 4
[3 marks]
For the circle (x−1)2+(y−2)2=9, state whether P(4,2) and Q(1,7) are inside, on, or outside the circle.
Question 5
[3 marks]
Explain why x2+y2+4x−2y+10=0 does not represent a real circle.
Question 6
[2 marks]
Find the points at which x2+y2=16 intersects the x-axis.
Section B
Topic practice
Build individual techniques
Question 7
[3 marks]
Find the points of intersection of (x−1)2+(y+2)2=20 and the line y=0.
Question 8
[4 marks]
A circle has equation x2+y2−4x−6y+9=0. Find its points of intersection with y=3.
Question 9
[4 marks]
Find the equation of the tangent to x2+y2=25 at P(3,4).
Question 10
[4 marks]
The line y=mx+5 is tangent to x2+y2=25. Find m.
Question 11
[5 marks]
Two identical circles with centres C1(1,2) and C2(7,2) touch externally. Find their radius, point of contact, and both equations.
Question 12
[5 marks]
Find the values of k for which y=x+k is tangent to (x−1)2+(y+2)2=8.
Question 13
[4 marks]
The diagram shows a circle with centre C(−1,4) and the point P(2,8) on its circumference. Find the equation of the tangent at P.
Question 14
[5 marks]
Find the exact coordinates of the points where the line y=2x+5 meets the circle (x+2)2+(y−1)2=25.
Section C
Exam-style and challenge
Combine skills, exam-style
Question 15
[5 marks]
The circle x2+y2−2x+4y+k=0 is tangent to the line y=3. Find k.
Question 16
[6 marks]
Find the exact points of intersection of (x−1)2+(y−1)2=10 and (x−5)2+(y−1)2=10.
Question 17
[5 marks]
A circle passes through A(0,0), B(4,0) and C(0,6). Find its equation.
Question 18
[6 marks]
From P(8,0), two tangents are drawn to x2+y2=16. Find their points of contact and equations.
Looking for the answers?
We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.
Are the answers to these worksheets available online?
No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.
Are these official SQA or Qualifications Scotland papers?
No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.
How should I use the three sections?
Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.
Can I print the worksheet?
Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.
