Determining the Area of a Triangle Formed by Axes and a Tangent Line to the Curve x2/3y2/3 a2/3

When dealing with mathematical curves and their tangents, understanding the geometric properties can be both intriguing and practical. In this article, we explore a specific curve and the triangle formed by its axes and a tangent line. The curve in question is defined by the equation x2/3y2/3 a2/3. We will derive the area of the triangle formed by the axes and the tangent line to this curve.

1. Evaluating the First-Order Derivative

To begin, we find the first-order derivative of the given curve. The equation is:

x2/3 y2/3 a2/3

We apply implicit differentiation to find the derivative dy/dx:

(2/3) x-1/3 y2/3 dx (2/3) x2/3 y-1/3 dy 0

Rearranging for dy/dx, we get:

dy/dx - (y/x)1/3

Thus, the slope of the tangent line at any point (p, q) on the curve is given by:

dy/dx - (q/p)1/3

2. Finding the Tangent Line Equation

Using the point-slope form of the equation for a tangent line, which is:

(y - q) / (x - p) - (q/p)1/3

Rearrange this equation to get:

(q/p)1/3 x - (q/p)1/3 p y - q

Simplifying, we find:

(q/p)1/3 x - (q/p)1/3 p y - q

(q/p)1/3 x (p/q)1/3 y a2/3

Here, we denote the constants as a2/3.

3. Finding the Intercepts

To find the intercepts of the tangent line with the x-axis and y-axis, we set:

x-intercept: y 0, then (q/p)1/3 x a2/3, hence x0 a2/3 p1/3

y-intercept: x 0, then (p/q)1/3 y a2/3, hence y0 a2/3 q1/3

4. Calculating the Area of the Triangle

The area of the triangle formed by the axes and the tangent line can be calculated using the formula for the area of a triangle:

A (1/2) x0 y0

Substituting the intercepts, we get:

A (1/2) a2/3 p1/3 a2/3 q1/3 (1/2) a4/3 p1/3 q1/3

Thus, the area of the triangle is:

A (1/2) a4/3 p1/3 q1/3

Conclusion

This exploration provides insight into the geometric properties of a specific curve and its tangent line. By understanding the first-order derivative and the intercepts, we can calculate the area of the triangle formed by the axes and the tangent line, which is given by the formula A (1/2) a4/3 p1/3 q1/3. Such an analysis is valuable in both theoretical and applied mathematics.

Keywords: tangent line, curve equation, triangle area