1. What is a Tangent Line to a Circle?

A tangent line to a circle is a straight line that touches the circle at exactly one point. At this point (called the point of tangency), the tangent line is perpendicular to the radius of the circle.

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( (h,k) \) — Center of the circle
• \( r \) — Radius of the circle
• \( (x_0,y_0) \) — Point of tangency
• \( m \) — Slope of the tangent line

Explanation: The slope of the tangent line is the negative reciprocal of the slope of the radius at the point of tangency.

3. Importance of Tangent Lines

Details: Tangent lines have applications in physics (e.g., instantaneous velocity), engineering (optimal curves), and computer graphics (smooth shading).

4. Using the Calculator

Tips: Enter the circle's center coordinates (h,k), its radius (r), and the point of tangency (x₀,y₀). The point must lie exactly on the circle.

5. Frequently Asked Questions (FAQ)

Q1: What if my point isn't on the circle?
A: The calculator assumes the point lies exactly on the circle. If it doesn't, there would be no tangent line at that point.

Q2: Can I find tangent lines to other shapes?
A: Yes, but the equations differ. This calculator is specifically for circles.

Q3: What does a vertical tangent line look like?
A: It would have an undefined slope and equation x = constant.

Q4: How is this related to derivatives?
A: In calculus, the derivative gives the slope of the tangent line to a curve at any point.

Q5: Can a line be tangent to a circle at two points?
A: No, by definition a tangent line touches at exactly one point. A line touching at two points is called a secant.