Library / Geometry, coordinates, surveying and trigonometry / GEO-37
Geometry, coordinates, surveying and trigonometry
Tangent line at a point on a circle
The constant C of the tangent Ax + By = C at the point (x_t, y_t) on a circle with centre (h, k), where A = x_t − h and B = y_t − k.
- Ref
- GEO-37
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- C [cm²]
At a point on a circle the radius is perpendicular to the tangent, so the radius vector (x_t − h, y_t − k) is the normal to the tangent. The tangent is A x + B y = C with A = x_t − h, B = y_t − k and C = A x_t + B y_t. The constant C is shown, and A and B are in the derived rows. Check that the point really is on the circle.
The constant C of the tangent Ax + By = C at the point (x_t, y_t) on a circle with centre (h, k), where A = x_t − h and B = y_t − k. The radius to the point is normal to the tangent. A and B are in the derived rows.
Calculator
- Coefficient A, x_t − h3 cm
- Coefficient B, y_t − k4 cm
- Radius to the point, √(A² + B²)5 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The tangent at (4, 6) to the circle centred at (1, 2) is 3x + 4y = 36, perpendicular to the 5 cm radius.
- A tangent point on the 3–4–5 radius, as in the standard example, gives the clean line 3x + 4y = 36.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| yt Point on the circle, y | −10 to 10 cm | 11.11 | 132 | 120.9 | +0.4 cm | +11.6 % |
| k Centre y | −10 to 10 cm | −12 | 108 | 120 | +0.4 cm | −6.67 % |
| xt Point on the circle, x | −10 to 10 cm | 23.78 | 134 | 110.2 | +0.4 cm | +8.22 % |
| h Centre x | −10 to 10 cm | 0 | 80 | 80 | +0.4 cm | −4.44 % |
Sweep of Point on the circle, x
| Point on the circle, x (cm) | Tangent line constant C (cm²) |
|---|---|
| −10 | 134 |
| −7.5 | 87.75 |
| −5 | 54 |
| −2.5 | 32.75 |
| 0 | 24 |
| 2.5 | 27.75 |
| 5 | 44 |
| 7.5 | 72.75 |
| 10 | 114 |
Sweep one variable
Variables and how each one contributes
Tangent line constant C
What the equation gives you. Everything else feeds it.
Centre x
The x-coordinate of the centre of the circle, in cm. Changes A and so tilts and shifts the line. Raising it by 0.5 (to 1.5) moves the result by −5.56 %, from 36 to 34 cm².
Centre y
The y-coordinate of the centre of the circle, in cm. Changes B and so tilts and shifts the line. Raising it by 0.5 (to 2.5) moves the result by −8.33 %, from 36 to 33 cm².
Point on the circle, x
The x-coordinate of the point of tangency, in cm. Changes both the slope and the position of the line. Raising it by 0.5 (to 4.5) moves the result by +10.4 %, from 36 to 39.75 cm².
Point on the circle, y
The y-coordinate of the point of tangency, in cm. Changes both the slope and the position of the line. Raising it by 0.5 (to 6.5) moves the result by +14.6 %, from 36 to 41.25 cm².
Worked example with the values above
- Normal vector (the radius): A = 4 − 1 = 3, B = 6 − 2 = 4.
- C = A x_t + B y_t = 3 × 4 + 4 × 6.
- C = 36 cm², so the tangent is 3x + 4y = 36.
Assumptions and limits
- Plane Cartesian coordinates, and a circle of positive radius unless stated.
- Angles in radians measured anticlockwise from the positive x-axis.
Common mistakes
- Forgetting the signs of D and E when reading the centre: it is (−D/2, −E/2).
- Not checking the discriminant before taking a square root.
- Using a point that is not actually on the circle in the tangent formula.
Statistics and uncertainty
These are exact results from the circle and line equations, but they need a root of a difference. Where the radicand is close to zero, as near tangency, small input errors can change the answer a lot or change whether a solution exists at all.
Where it is used
- Circle fitting, wheel and cam geometry, hole patterns and clearance checks.
- Finding where a path, ray or cut meets a circular boundary.
Maths you need
- Equation of a circle: (x − h)² + (y − k)² = r²
- Quadratic discriminant: b² − 4ac decides the number of roots
- Gradient of a normal: perpendicular to the tangent