Circles: tangents, chords and lines

Three properties

• The angle in a semicircle is 90°: if AB is a diameter and C is on the circle, angle ACB = 90°.
• The perpendicular from the centre to a chord bisects the chord.
• A tangent is perpendicular to the radius at the point of contact.

Tangent at a point

Find the gradient of the radius from the centre to the point. The tangent gradient is its negative reciprocal. Then use y - y_1 = m(x - x_1).

Lines meeting circles

Substitute the line into the circle to get a quadratic. b² - 4ac > 0: two points; = 0: the line is a tangent; < 0: they do not meet.

Worked example

Find the tangent to x² + y² = 25 at (3, 4).

1. The centre is (0, 0), so the radius has gradient 4/3
2. The tangent gradient is -3/4
3. y - 4 = -3/4 (x - 3)
4. 4y - 16 = -3x + 9

Answer: 3x + 4y = 25

Check yourself

1. Find the gradient of the tangent to (x - 1)² + (y - 2)² = 25 at the point (4, 6). Give your answer as a fraction in its simplest form.
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   -3/4. The radius has gradient (6 - 2)/(4 - 1) = 4/3, so the tangent gradient is -3/4.

2. A chord is 6 cm from the centre of a circle of radius 10 cm. Find the length of the chord.
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   16 cm. Half the chord is sqrt(100 - 36) = 8, so the chord is 16 cm.

3. AB is a diameter of a circle and C is another point on the circle. What is angle ACB?
   • 180°
   • 45°
   • 90°
   • 60°
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   90°. The angle in a semicircle is a right angle.

4. How many points do the line y = x + 5 and the circle x² + y² = 4 have in common?
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   0. x² + (x + 5)² = 4 gives 2x² + 10x + 21 = 0. b² - 4ac = 100 - 168 = -68 < 0, so they do not meet.

5. The line y = k is a tangent to (x - 2)² + (y - 3)² = 16. Find the positive value of k.
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   7. The radius is 4, so horizontal tangents are 4 above and below the centre: y = 7 or y = -1. The positive value is 7.

6. A(-1, 2) and B(5, 10) are the ends of a diameter. Find the radius of the circle.
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   5. AB = sqrt(36 + 64) = 10, so the radius is 5.

7. P is the point (7, 1). Find the length of the tangent from P to the circle x² + y² = 25.
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   5. The distance from the centre to P is sqrt(50). The tangent and radius meet at 90°, so the tangent length is sqrt(50 - 25) = 5.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.