Understanding the Angle of Intersection of Two Curves Touching Each Other

When two curves touch each other at a specific point, they share that point without crossing through it. This unique geometric condition naturally leads to an interesting question: what is the angle of intersection between two curves that touch each other?

The Concept of Tangents

To clarify this concept, let's delve into the notion of tangents. At the point where two curves touch, their tangent lines are not just touching but are coincident. This means that the derivatives of the functions representing the curves are equal at that point.

Consider the derivative of a function as the slope of the tangent line at a particular point. When two curves touch, their slopes (derivatives) are identical at the point of contact. Consequently, the tangent lines to both curves at this point are the same.

Defining the Angle of Intersection

The angle of intersection between two curves can be defined as the angle between their tangent lines at the point of intersection. However, when the curves touch rather than cross, the tangent lines coincide, simplifying the angle of intersection to zero degrees.

Mathematical Approach

To find the angle of intersection, one must first determine the slopes of the tangents at the point of contact. This involves differentiating the equations of the curves and evaluating the derivatives at the point of intersection.

Differentiation and Slopes

Suppose we have two curves y f(x) and y g(x). To find the tangents at a point where they touch, we calculate the derivatives f'(x) and g'(x), which represent the slopes of the respective tangents.

Substitute the coordinates of the point of contact into the derivatives to find the slopes of the tangents. The angle θ between the tangents can be calculated using the tangent formula:

If the curves touch at the point, the slopes are equal (f'(x) g'(x)), causing the numerator to be zero and thereby the angle between the tangents to be zero.

Visualizing the Intersection

To better understand the concept, visualize two curves touching at a point. Imagine drawing tangents to each curve at that point. If the curves touch, the tangents will lie on top of each other, forming an angle of 0°.

Example Calculation

Consider two curves defined by:
y x^2 3x 2 and y 2x - 1
We find the point of tangency by setting the derivatives equal:
2x 3 2
Solving for x gives x -1/2. Substituting this back into either equation gives the corresponding y-value, (y 1.5).

At x -1/2, both tangents have the same slope of 2, confirming that the angle between them is 0°.

Conclusion

In summary, when two curves touch each other at a point, they share that point without crossing, making the angle of intersection between their tangent lines at the point of contact precisely 0°. This unique condition highlights the importance of the derivative in understanding the geometric properties of curves.