## Point of Tangency

### The point of tangency is a point on a curve or a circle where a straight line, called the tangent line, touches the curve or circle at only one point

The point of tangency is a point on a curve or a circle where a straight line, called the tangent line, touches the curve or circle at only one point. At the point of tangency, the tangent line is perfectly aligned with the curve, meaning that the two have the same slope at that point.

In geometry, the point of tangency is important for various reasons. For instance, when studying circles, the point of tangency helps us determine the relationship between the circle and a tangent line. The tangent line is perpendicular to the radius of the circle that joins the center of the circle to the point of tangency. This relationship allows us to calculate angles and lengths within the circle.

In calculus, the concept of the point of tangency is central to differentiation. When finding the derivative of a function, the derivative at a particular point corresponds to the slope of the tangent line to the graph of the function at that point.

Understanding the point of tangency is crucial for various applications, including physics, engineering, and optimization problems, where finding the maximum or minimum of a function requires analyzing the behavior of the tangent line at critical points.

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