Calculation and Analysis of the Area of a Triangle Formed by a Tangent to the Curve (f(x) frac{1}{x})

Understanding the properties of mathematical functions and their tangents can lead to fascinating geometric insights. One such exploration involves examining the area of a triangle formed by a tangent line to the curve (f(x) frac{1}{x}) and the coordinate axes. This article delves into this intriguing problem and its implications for further mathematical exploration.

Introduction

The curve (f(x) frac{1}{x}) is a hyperbola, symmetric about the origin and defined on the set of all real numbers except zero. The tangent line to this curve at any point (x a) intersects the coordinate axes, forming a right triangle. The primary goal of this article is to derive the area of such a triangle and to verify if this area remains constant for all points on the curve.

Derivation of the Tangent Line Equation

Let's start by finding the equation of the tangent line to (f(x) frac{1}{x}) at the point ((a, f(a))).

Choose a point on the curve: Let's select (a) such that (f(a) frac{1}{a}).

At (x a), the function value is (f(a) frac{1}{a}).

Derivative Calculation

The derivative of (f(x)) is:

[f'(x) -frac{1}{x^2}]

Thus, at (x a), the derivative is (f'(a) -frac{1}{a^2}).

Tangent Line Equation

The equation of the tangent line at the point ((a, frac{1}{a})) is given by:

[y - frac{1}{a} -frac{1}{a^2}(x - a)]

Rearranging this, we get:

[y -frac{1}{a^2}x frac{1}{a} frac{1}{a}]

[y -frac{1}{a^2}x frac{2}{a}]

Finding the Intercepts

The intercepts of the tangent line with the coordinate axes are crucial for calculating the area of the triangle.

Set (y 0):

[0 -frac{1}{a^2}x frac{2}{a}]

[-frac{1}{a^2}x -frac{2}{a}]

[frac{1}{a^2}x frac{2}{a}]

[x 2a]

The X-intercept is at ((2a, 0)).

Set (x 0):

[y frac{2}{a}]

The Y-intercept is at ((0, frac{2}{a})).

Calculating the Area of the Triangle

The vertices of the triangle are at ((0, 0)), ((2a, 0)), and ((0, frac{2}{a})).

The base of the triangle is (2a), and the height is (frac{2}{a}).

The area (A) of the triangle can be calculated using the formula:

[A frac{1}{2} times text{base} times text{height} frac{1}{2} times 2a times frac{2}{a} 2]

Conclusion

Regardless of the point (a) chosen on the curve (f(x) frac{1}{x}), the area of the triangle formed by the tangent line, the x-axis, and the y-axis is always 2. This result highlights a remarkable property of the curve and demonstrates the stability of certain geometric properties despite the varying nature of the function's behavior.

It is important to note that (f(x)) and its derivative are undefined at (x 0). As such, no triangle can be formed with a tangent at (x 0) or any point within the set of (x 0).

This constant area property of the triangle offers a unique perspective on the relationship between the function and its geometric interpretations, making it a valuable topic for further exploration in both theoretical and applied mathematics.