Calculator Inputs
Example Data Table
| Circle | Line | Expected Type | Example Point Data |
|---|---|---|---|
| Center (0, 0), radius 5 | x + y - 5 = 0 | Secant | (0, 5) and (5, 0) |
| Center (0, 0), radius 5 | x - 5 = 0 | Tangent | (5, 0) |
| Center (0, 0), radius 5 | x - 6 = 0 | No intersection | No real point |
Formula Used
The calculator models a circle as (x - h)² + (y - k)² = r² and a line as ax + by + c = 0.
- Distance from center to line:
d = |ah + bk + c| / √(a² + b²) - Perpendicular foot:
F = (h, k) - ((ah + bk + c)/(a² + b²))(a, b) - Decision rule: secant when
d < r, tangent whend = r, no real hit whend > r - Intersection points:
P = F ± t(-b, a)/√(a² + b²), wheret = √(r² - d²) - Chord length:
2√(r² - d²)for secant cases
The verification fields report how close each computed point comes to satisfying both the circle equation and the line equation.
How to Use This Calculator
- Enter the circle center coordinates and the radius.
- Select either standard-form line input or two-point line input.
- Provide the matching line values. Use the segment option only for two-point mode.
- Choose the output precision and press Calculate Intersection.
- Review the result table above the form, then export it as CSV or PDF if needed.
FAQs
1. What does a secant result mean?
A secant result means the line crosses the circle at two distinct real points. The calculator reports both coordinates, the perpendicular foot, and the chord length between the two contact points.
2. What does a tangent result mean?
A tangent result means the line touches the circle at exactly one point. In that case, the perpendicular foot from the center to the line is also the contact point.
3. Why can the calculator show no intersection?
No intersection appears when the shortest distance from the circle center to the line is greater than the radius. The line stays completely outside the circle, so there are no real meeting points.
4. Which line format is better to use?
Use standard form when you already know coefficients a, b, and c. Use two-point mode when you want the calculator to derive the line from coordinates automatically.
5. What does the segment filter do?
The segment filter checks whether computed line-circle intersection points actually lie between the two chosen endpoints. This is useful when your input represents a finite edge rather than an infinite line.
6. Why are there circle and line check values?
These values verify numerical accuracy. A value close to zero means the reported point satisfies the original equations very closely, which helps confirm a stable result.
7. Can the calculator handle vertical lines?
Yes. Vertical lines work naturally in standard form and also in two-point mode. The underlying geometry uses vector methods, so no slope division is required.
8. What units should I use?
Use one consistent unit system for every coordinate and the radius. The calculator does not convert units, so mixed units would make the output physically meaningless.