Higher Maths · Applications
The Circle | Higher Maths
The Circle is one of the most reliable mark-grabbers in Higher Maths. Tangents to circles and intersections with straight lines are tested almost every year. Confidence with completing the square is essential.
Circle equations
Circle centre (a, b) radius r: (x − a)2 + (y − b)2 = r2
General form: x2 + y2 + 2gx + 2fy + c = 0
Centre: (−g, −f)
Radius: r = √(g2 + f2 − c)
A real circle exists only when g2 + f2 − c > 0.
Worked example
Worked example — Tangent to a circle
Problem: Find the equation of the tangent to the circle (x − 2)2 + (y + 1)2 = 25 at the point P(5, 3).
- Identify the centre.Centre C = (2, −1).
- Find the gradient of the radius CP.mCP = (3 − (−1)) / (5 − 2) = 4/3
- Tangent gradient is the negative reciprocal.mtangent = −3/4
- Equation of tangent through (5, 3).y − 3 = (−3/4)(x − 5) ⇒ 4y + 3x = 27
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Find the centre and radius of x2 + y2 − 6x + 4y − 12 = 0.
Show answer
Centre (3, −2), radius 5 - Does the line y = x + 1 cut the circle x2 + y2 = 25? If so, find the points.
Show answer
Substitute: x2 + (x+1)2 = 25 ⇒ 2x2 + 2x − 24 = 0 ⇒ x = 3 or −4. Points: (3, 4) and (−4, −3) - Find the equation of the circle with centre (−1, 4) and radius √10.
Show answer
(x + 1)2 + (y − 4)2 = 10 - Show that the line y = 2x + 5 is a tangent to x2 + y2 = 5.
Show answer
Substituting gives 5x2 + 20x + 20 = 0 ⇒ x2 + 4x + 4 = 0 ⇒ (x + 2)2 = 0. Repeated root means tangent. - Find the value(s) of k such that the line y = x + k is a tangent to x2 + y2 = 8.
Show answer
2x2 + 2kx + k2 − 8 = 0. Discriminant 4k2 − 8(k2 − 8) = 0 ⇒ k2 = 16. k = ±4
Common mistakes
Common mistakes & how to avoid them
- Mistaking the centre as (g, f) instead of (−g, −f) in the general form.
- Forgetting to check that the radius is real (g2 + f2 − c > 0) before claiming a circle exists.
- Using the gradient of the radius for the tangent — they are perpendicular.
Frequently asked questions
Is the circle equation on the SQA formula sheet?
Yes — both forms (centre-radius and general) are on the Higher formula sheet. But you must still know how to extract the centre and radius from the general form.
How do I prove a line is a tangent without using the gradient method?
Substitute the line into the circle. Solve the resulting quadratic. If the discriminant is zero, there is exactly one intersection — meaning the line is a tangent.
Can a circle pass through a given set of points?
Three non-collinear points uniquely define a circle. You can find its equation by setting up three equations in g, f and c and solving simultaneously.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
Need one-to-one help with The Circle?
Our Glasgow Higher Maths specialists run 1-hour one-to-one sessions and 80-minute group classes — online or at our Pollokshields Tuition Centre.
Ready to enrol?
Get matched with a Glasgow specialist
Tell us the subject and level — we’ll come back the same day with availability and a clear quote.