Tangent Line to a Circle

Find the equation of the line tangent to a circle at a given point on its circumference.

Enter the circle's center coordinates (h, k), its radius (r), and the coordinates of a point (x, y) on the circle to find the tangent line equation.

Practical Examples

Explore these examples to see how the calculator works in different scenarios.

Standard Circle at Origin

example1

A circle centered at the origin (0,0) with a radius of 5. Find the tangent at point (3,4).

Center Coordinates: h: 0, k: 0

Radius: r: 5

Point Coordinates: (3, 4)

Offset Circle

example2

A circle centered at (2,-1) with a radius of 10. Find the tangent at point (8,7).

Center Coordinates: h: 2, k: -1

Radius: r: 10

Point Coordinates: (8, 7)

Horizontal Tangent

example3

A circle centered at (1,1) with a radius of 3. Find the tangent at the top point (1,4).

Center Coordinates: h: 1, k: 1

Radius: r: 3

Point Coordinates: (1, 4)

Vertical Tangent

example4

A circle centered at (-2,3) with a radius of 4. Find the tangent at the rightmost point (2,3).

Center Coordinates: h: -2, k: 3

Radius: r: 4

Point Coordinates: (2, 3)

Other Titles
Understanding Tangent Lines: A Comprehensive Guide
An in-depth look at the geometry of circles and tangent lines, from core principles to mathematical derivations and practical applications.

What is a Tangent to a Circle?

  • Defining the tangent line and point of tangency.
  • The fundamental relationship between a tangent and the circle's radius.
  • Contrasting a tangent with a secant and a chord.
In Euclidean geometry, a tangent line to a circle is a straight line that touches the circle at exactly one point, without entering the circle's interior. This single point where the line and circle meet is known as the point of tangency. The concept is a cornerstone of coordinate geometry and has wide-ranging applications in fields like physics (e.g., describing the instantaneous velocity of a point moving in a circular path) and computer graphics (e.g., calculating light and shadow).
The Tangent-Radius Theorem
The most critical property of a tangent line is its relationship with the radius of the circle at the point of tangency. The Tangent-Radius Theorem states that the radius drawn to the point of tangency is always perpendicular (at a 90-degree angle) to the tangent line. This perpendicular relationship is the key to deriving the equation of the tangent line.
Tangent vs. Secant
It's important to distinguish a tangent from a secant. While a tangent touches the circle at a single point, a secant is a line that intersects the circle at two distinct points. A chord is the line segment connecting these two intersection points of a secant.

Key Concepts

  • A line touching a bicycle wheel at one point is a tangent.
  • The radius to the point of tangency is the shortest distance from the circle's center to the tangent line.
  • A secant line goes through the circle, while a tangent line skims its edge.

Step-by-Step Guide to Using the Calculator

  • Entering the circle's defining parameters: center and radius.
  • Specifying the exact point of tangency on the circle.
  • Interpreting the different forms of the resulting line equation.
Our calculator simplifies the process of finding the tangent line equation. Follow these steps to get an accurate result:
1. Input Circle Information
Start by defining the circle. You need to provide the coordinates of its center (h, k) and its radius (r). The radius must be a positive value.
2. Input the Point of Tangency
Next, provide the coordinates (x₁, y₁) of the point where the tangent line touches the circle. For the calculation to be valid, this point must lie on the circle's circumference. Our calculator automatically verifies this for you.
3. Calculate and Interpret the Results
Press the 'Calculate' button. The tool will display the tangent line's equation in two common formats: the General Form (Ax + By + C = 0) and the Slope-Intercept Form (y = mx + c). For vertical lines, the slope-intercept form is not applicable, and this will be indicated.

Input Fields

  • Circle Center (h, k): The 'anchor' point of the circle.
  • Radius (r): The size of the circle.
  • Point of Tangency (x₁, y₁): The point 'on' the circle's edge.

Mathematical Derivation and Formulas

  • Using the point-slope form as the foundation.
  • Calculating the slope of the radius and the tangent.
  • Handling special cases like horizontal and vertical tangents.
The calculation is based on the Tangent-Radius Theorem. The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
Deriving the Slope
1. First, we find the slope of the radius connecting the center (h, k) to the point of tangency (x₁, y₁). This slope, m_radius, is given by (y₁ - k) / (x₁ - h).
2. Since the tangent is perpendicular to the radius, its slope, mtangent, is the negative reciprocal of the radius's slope. So, mtangent = -1 / m_radius = -(x₁ - h) / (y₁ - k).
Forming the Equation
With the slope of the tangent and the point of tangency (x₁, y₁), we can use the point-slope form of a line, y - y₁ = m(x - x₁), to write the equation: y - y₁ = (-(x₁ - h) / (y₁ - k)) * (x - x₁). This equation can then be rearranged into the general and slope-intercept forms.
Special Cases
If y₁ - k = 0, the radius is horizontal, making the tangent line vertical. Its equation is simply x = x₁. If x₁ - h = 0, the radius is vertical, making the tangent line horizontal, with the equation y = y₁.

Formulas Used

  • Circle Equation: (x - h)² + (y - k)² = r²
  • Radius Slope: m_radius = (y₁ - k) / (x₁ - h)
  • Tangent Slope: m_tangent = -(x₁ - h) / (y₁ - k)
  • Line Equation: y - y₁ = m(x - x₁)

Real-World Applications

  • Applications in physics and engineering.
  • Use cases in computer graphics and animation.
  • Importance in architecture and design.
The concept of a tangent to a circle is not just a theoretical exercise; it has numerous practical applications.
Physics and Engineering
In mechanics, the velocity of an object moving in a circular path is always tangent to the circle at any given point. This is why if you swing an object on a string and let go, it flies off in a straight line tangent to its circular path. It's also used in designing gears and pulley systems to ensure smooth transmission of motion.
Computer Graphics
In 2D and 3D graphics, tangents are crucial for creating smooth curves (using splines), calculating lighting effects, and determining how objects should collide or interact realistically.
Architecture and Navigation
Architects use tangents to design curved structures like domes and arches. In navigation and surveying, tangent lines are used in sightline calculations and for mapping.

Practical Scenarios

  • Designing a belt drive system with two pulleys.
  • Calculating the path of a satellite leaving orbit.
  • Creating smooth, curved roads in a city plan.

Common Questions and Pitfalls

  • Verifying the point is truly on the circle.
  • Handling the undefined slope of a vertical tangent.
  • Understanding the difference between circle forms.
When working with tangents, a few common issues can arise. Understanding them helps in avoiding errors.
Is the Point on the Circle?
The most common mistake is trying to calculate a tangent for a point that is not on the circle. The distance from the center (h, k) to the point (x₁, y₁) must equal the radius r. If (x₁ - h)² + (y₁ - k)² ≠ r², the point is either inside or outside the circle, and this specific tangent formula does not apply. Our calculator checks this condition automatically.
Vertical Tangents
A vertical tangent line has an undefined slope. This occurs when the point of tangency is directly to the left or right of the circle's center (i.e., y₁ = k). In this case, the slope-intercept form y = mx + c is not applicable. The equation is simply x = x₁, a vertical line passing through the tangent point.
Numerical Precision
Due to floating-point arithmetic in computers, checking if a point is on the circle might involve a small tolerance. For example, instead of checking if (x₁ - h)² + (y₁ - k)² equals r² exactly, we check if the difference is very close to zero.

Points to Remember

  • Always double-check that your point satisfies the circle's equation.
  • A vertical tangent has an 'undefined' slope, not a slope of zero.
  • A horizontal tangent has a slope of zero.