GCSE-ready- Home
- Lessons
- GCSE & IGCSE Mathematics
- Equation of a circle and tangents
Equation of a circle and tangents
A circle centred at the origin has a neat equation from Pythagoras. Tangents meet the radius at right angles.
The equation
A circle with centre (0, 0) and radius r: x2 + y2 = r2
x2 + y2 = 25 has radius 5.
x2 + y2 = 25 has radius 5.
Tangents
A tangent touches the circle at one point and is perpendicular to the radius there.
Gradient of radius to (3, 4): 43. Tangent gradient: −34.
Gradient of radius to (3, 4): 43. Tangent gradient: −34.
Equation of a tangent
Tangent at (3, 4) on x2 + y2 = 25:
m = −34. 4 = −34 × 3 + c, so c = 254.
y = −34x + 254
m = −34. 4 = −34 × 3 + c, so c = 254.
y = −34x + 254
Is the point (1, 3) on the circle x2 + y2 = 10?
- 12 + 32 = 1 + 9 = 10
- This equals 10
Answer: Yes
x2 + y2 = r2 is a circle centred at the origin. A tangent is perpendicular to the radius: find the radius gradient, flip it and change the sign.
Check you have got it
Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.