Problems Based on Angle of Intersection of Two Curves

The angle of intersection between two curves at a point is defined as the angle between the tangents to the curves at their point of intersection. This concept is particularly important in calculus and analytical geometry. To find the angle of intersection, we need to understand the slopes of the tangents to the curves at the point of intersection.

Understanding Slopes and Tangents

The slope of a tangent to a curve at a given point is the derivative of the curve's equation at that point. If we have two curves, $y = f(x)$ and $y = g(x)$, and they intersect at a point $(x_0, y_0)$, the slopes of the tangents to these curves at $(x_0, y_0)$ are $f'(x_0)$ and $g'(x_0)$, respectively.

Angle of Intersection Formula

The angle of intersection, $\theta$, between two curves can be found using the formula:

[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| ]

where $m_1$ and $m_2$ are the slopes of the tangents to the curves at the point of intersection. The absolute value is used to ensure that the angle is non-negative, as the tangent function can yield negative values.

Table of Differences and Important Points

Aspect	Description
Slope of Tangent	The slope of the tangent to a curve at a point is the derivative of the curve at that point.
Angle of Intersection	The angle between the tangents to two curves at their point of intersection.
Formula	$\tan(\theta) = \left
Acute or Obtuse	The angle calculated using the formula is always the acute angle. To find the obtuse angle, subtract the acute angle from $\pi$.

Examples

Example 1: Intersection of Two Lines

Consider the lines $y = 2x + 3$ and $y = -x + 1$. The slopes of these lines are $m_1 = 2$ and $m_2 = -1$, respectively.

Using the formula:

[ \tan(\theta) = \left| \frac{2 - (-1)}{1 + 2(-1)} \right| = \left| \frac{3}{-1} \right| = 3 ]

Thus, $\theta = \arctan(3)$, which gives us the acute angle of intersection.

Example 2: Intersection of a Line and a Parabola

Consider the line $y = x - 2$ and the parabola $y = x^2$. The slope of the line is $m_1 = 1$. To find the slope of the tangent to the parabola at the point of intersection, we first need to find the derivative of the parabola, which is $2x$. If they intersect at $x = 1$, the slope of the tangent to the parabola at that point is $m_2 = 2(1) = 2$.

Using the formula:

[ \tan(\theta) = \left| \frac{1 - 2}{1 + 1(2)} \right| = \left| \frac{-1}{3} \right| = \frac{1}{3} ]

Thus, $\theta = \arctan\left(\frac{1}{3}\right)$, which gives us the acute angle of intersection.

Example 3: Intersection of Two Parabolas

Consider the parabolas $y = x^2$ and $y = -x^2 + 4x - 3$. To find the angle of intersection, we need to find the slopes of the tangents at the point of intersection. The derivatives are $2x$ and $-2x + 4$, respectively. Suppose they intersect at $x = 1$, then the slopes are $m_1 = 2(1) = 2$ and $m_2 = -2(1) + 4 = 2$.

Since the slopes are equal, the tangents are parallel, and the angle of intersection is $\theta = 0$.

Conclusion

The angle of intersection of two curves is an important concept in calculus and geometry. By understanding the slopes of tangents and using the formula provided, one can calculate the angle of intersection between any two curves. It is important to remember that the formula yields the acute angle, and if an obtuse angle is required, one must subtract the acute angle from $\pi$. Practice with various types of curves is essential to master this concept.