Two circle facts create right angles you can use: a tangent is perpendicular to the radius at the point of contact, and the perpendicular from the centre to a chord bisects the chord. Add the discriminant, which tells you whether a line cuts a circle, touches it or misses it, and you can solve almost any line-and-circle question.
Key facts for tangents and chords
- Tangent gradient = −1 / (gradient of the radius to the point of contact)
- Perpendicular distance d, half-chord and radius: d² + (½ chord)² = r²
- Tangent length from an external point: √(d² − r²), where d is the distance to the centre
Line and circle: substitute and use the discriminant
- b² − 4ac > 0: two points, so the line is a chord
- b² − 4ac = 0: one point, so the line is a tangent
- b² − 4ac < 0: the line misses the circle
Why is a tangent perpendicular to the radius?
The tangent at a point on a circle touches it without crossing, and it always meets the radius to that point at 90°. That gives the tangent's gradient: find the gradient of the radius from the centre to the point, then take the negative reciprocal. With one point and a gradient, the equation follows.
The normal at that point is perpendicular to the tangent, so it is the radius extended and always passes through the centre. Use the gradient from the centre, not from the origin.
The perpendicular from the centre bisects a chord
Drop a perpendicular from the centre to a chord and a right-angled triangle appears, with the radius as the hypotenuse, the perpendicular distance as one side and half the chord as the other. Given any two of the three, Pythagoras gives the third.
The triangle uses half the chord. Forgetting to halve, or to double back at the end, is the most common slip.
Does a line cut, touch or miss a circle?
Make y the subject of the line and substitute it into the circle's equation. You get a quadratic in x, and its discriminant answers the question: positive means two intersection points, zero means the line is a tangent, negative means no intersection.
The zero case is the useful one. A tangent gives a repeated root, so setting b² − 4ac = 0 is how you prove a line is a tangent, or find an unknown constant that makes it one.
The length of a tangent from a point outside the circle
From an external point, the centre, the point and the point of contact form a right-angled triangle, with the right angle where the tangent meets the radius. The hypotenuse is the distance from the external point to the centre, so the tangent length is √(d² − r²).
If d is less than r, the point is inside the circle and no tangent exists, which is a useful sanity check.
Choosing the right property
Circle questions often chain several steps. Let the wording point you to the tool: “tangent” means perpendicular to the radius; “chord” means the perpendicular from the centre bisects it; “diameter” means the angle in a semicircle is 90°; “how many points” means substitute and use the discriminant. A quick sketch with the centre marked makes the right angle obvious.
Worked examples
Example 1. A circle has radius 5. A chord AB is at a perpendicular distance 3 from the centre. Find the length of AB.
- The perpendicular from the centre O meets AB at its midpoint M, so triangle OMB is right-angled at M.
- OM = 3 and OB = 5 (a radius, the hypotenuse).
- MB² = 25 − 9 = 16, so MB = 4.
- MB is half the chord, so AB = 8.
Answer: AB = 8.
Marks: M1 for the right-angled triangle, M1 for Pythagoras, A1 for doubling.
Example 2. A circle has centre (3, 1) and radius 5. Find the length of the tangent from T(9, 9), as an exact surd.
- Distance from T to the centre: √(6² + 8²) = 10.
- 10 > 5, so T is outside and a tangent exists.
- Tangent² = d² − r² = 100 − 25 = 75.
- √75 = 5√3.
Answer: 5√3.
Marks: M1 for d, M1 for Pythagoras, A1 for the exact surd.
Example 3. The circle S has centre (−1, 4) and passes through P(3, 7). (a) Find r². (b) Find where the tangent at P crosses the x-axis.
- (a) r² = (3 − (−1))² + (7 − 4)² = 16 + 9 = 25.
- (b) Gradient of the radius SP: 3/4, so the tangent's gradient is −4/3.
- Tangent through P: y − 7 = −4/3 (x − 3), which gives 4x + 3y − 33 = 0.
- On the x-axis y = 0: 4x = 33, so x = 33/4.
Answer: r² = 25; the tangent crosses the x-axis at (33/4, 0).
Marks: B1 for r², M1 for the tangent, A1 for the crossing point.
Common mistakes
- Using the gradient from the origin. The radius runs from the centre of the circle to the point of contact.
- Forgetting to halve or double the chord. The right-angled triangle uses half the chord. Double your answer for the full length.
- Substituting the circle into the line. Make y the subject of the line and substitute into the circle, so you get a single quadratic in x.
- Using the radius as the hypotenuse for a tangent length. For a tangent from an external point, the hypotenuse is the distance to the centre.
- Stopping at the discriminant. If the question also asks for the points, solve the quadratic and find both coordinates.
How tangents and chords are examined
Tangents and chords are Pure 2 content in Edexcel International A-Level (paper WMA12), in the coordinate geometry chapter, and first-year Pure in Edexcel A-Level (9MA0). Questions commonly ask you to:
- find the equation of a tangent or normal at a point
- find a chord length or the distance of a chord from the centre
- show that a line is a tangent, or find the values of k that make it one
- find the length of a tangent from an external point
Full multi-part questions with mark schemes are in the Pure 2 coordinate geometry exam questions. Practise further with the coordinate geometry worksheet or the coordinate geometry revision engine.
You need the equation of a circle and the discriminant from quadratic functions for this course.
Frequently asked questions
How do you find the equation of a tangent to a circle?
Find the gradient of the radius from the centre to the point, take the negative reciprocal to get the tangent's gradient, then use y − y₁ = m(x − x₁).
How do you prove a line is a tangent to a circle?
Substitute the line into the circle to get a quadratic and show that its discriminant is zero. A repeated root means the line touches the circle at exactly one point.
What is the difference between a tangent and a normal?
A tangent touches the circle and is perpendicular to the radius. A normal is perpendicular to the tangent, so it lies along the radius and passes through the centre.