Tangent Lines to Circles

This page focuses on finding the equation of a tangent line to a circle, a key concept in analytic geometry. The main example presented is finding the tangent line to the circle x² + y² = 13 at the point (-3, 2).

Definition: A tangent line to a circle is a line that touches the circle at exactly one point, called the point of tangency.

The process of finding the tangent line equation involves several steps:

1. Identify the point of tangency (-3, 2) and the circle equation x² + y² = 13.
2. Calculate the slope of the radius at the point of tangency, which is perpendicular to the tangent line.
3. Use the perpendicular slope to find the slope of the tangent line.
4. Apply the point-slope form of a line equation to derive the tangent line equation.

Example: For the circle x² + y² = 13 and point of tangency (-3, 2), the tangent line equation is derived as y = (3/4)x + 13/4.

The page includes a diagram illustrating the circle, the point of tangency, and the tangent line, which helps visualize the geometric relationship.

Highlight: Understanding how to find the equation of tangent to a circle from an external point is crucial for solving more advanced problems in geometry and calculus.

This example demonstrates the practical application of circle equations and line equations in solving geometric problems. It showcases how analytical methods can be used to describe geometric relationships precisely.