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Angle of Intersection between Two Curves: Formula, Examples

Angle of Intersection between Two Curves: Formula, Examples

Edited By Komal Miglani | Updated on Jul 02, 2025 07:51 PM IST

The angle of Intersection is an important concept in calculus. It is used to find out the angle between the curves. The tangent line to the curve is a straight line that touches a curve at a single point without crossing it at that point. These concepts of Angle of Intersection between two curves have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

This Story also Contains
  1. The angle of Intersection of Two Curves
  2. Orthogonal Curves
  3. Solved Examples Based On Angle of Intersection of Two Curves:
Angle of Intersection between Two Curves: Formula, Examples
Angle of Intersection between Two Curves: Formula, Examples

In this article, we will cover the concept of the Angle of Intersection of two Curves. This topic falls under the broader category of Calculus, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of six questions have been asked on this topic in JEE Main from 2013 to 2023, including one in 2013, one in 2018, one in 2019, and three in 2021.

Background wave

What is the Angle of Intersection?

The angle of intersection of two curves is defined as the angle between the tangents to the two curves at their point of intersection
Let C1 and C2 be two curves having equations y=f(x) and y=g(x), respectively.
Let PT1 and PT2 be two tangents to the curves C1 and C2 at their point of intersection.
Let θ be the angle between the two tangents PT1 and PT2 and θ1 and do tangents make the angles with the positive direction of the X -axis in the anti-clockwise sense.

Then

m2=tanθ2=(dy dx)C2


From the figure it follows, θ=θ2θ1

tanθ=tan(θ2θ1)=tanθ2tanθ21+tanθ2tanθ1⇒=tanθ|(dy dx)C1(dy dx)C21+(dy dx)C1(dy dx)C2|

The intersection of these curves is defined as the acute angle between the tangents.

The angle of Intersection of Two Curves

Let y = f (x) and y = g (x) be two curves intersecting at a point P(x0, y0) . Then the angle of intersection of two curves is defined as the angle between the tangent to the two curves at the point of intersection.

Let PT1 and PT2 be tangents to the curve y=f(x) and y=g(x) at their point of intersection.
Let Θ be the angle between two tangents PT1 and PT2,Θ1 and Θ2 are angles made by tangents PT1 and PT2 with the positive direction of x-axis, then

m1=tanθ1=(ddx(f(x)))(x0,y0)m2=tanθ2=(ddx(g(x)))(x0,y0)

from the figure θ=θ1θ2

tanθ=tan(θ1θ2)=tanθ1tanθ21+tanθ1tanθ2tanθ=|(ddx(f(x)))(x0,yo)(ddx(g(x)))(x0,yo)1+(ddx(f(x)))(x0,yo)(ddx(g(x)))(x0,y0)|

Orthogonal Curves

If the angle of the intersection of two curves is a right angle then two curves are called orthogonal curves.
In this case, tanθ=90

(ddx(f(x)))(x0,y0)(ddx(g(x)))(x0,y0)=1

this is also the condition for two curves to be orthogonal.

Condition for two curves to touch each other

(ddx(f(x)))(x0,y0)=(ddx(g(x)))(x0,y0)

Recommended Video Based on Angle of Intersection of Two Curves:


Solved Examples Based On Angle of Intersection of Two Curves:

Example 1: If the curves y2=6x,9x2+by2=16 intersect each other at right angles, then the value of b is :
[JEE Main 2028]
1) 9/2
2) 6
3) 7/2
4) 4

Solution
As we have learned
Condition of Orthogonality -
Two curves intersect each other orthogonally if the tangents to each of them subtend a right angle at the point of intersection of two curves:

m1×m2=1

y2=6x9x2+6bx=16


Slope of tangent of first curve

2ydydx=6dydx=62ym1=62y


Slope of tangent of second curve

18x+2bydydx=0dydx=9xby9xby=m2

So m1×m2=1

m1m2=1(62y)(bx9y)=127x=by227x=b(6x)b=276=92

Example 2: limy01+1+y42y4
[JEE Main 2019]
1) exists and equals 142
2) exists and equals 122(2+1)
3) exists and equals 122
4) does not exist|

Solution
Angle of intersection of two curves -
The angle of intersection of two curves is the angle subtended between the tangents at their point of intersection Let m1 \& m2 are two slope of tangents at intersection point of two curves then

tanθ=[m1m2]1+m1m2

where θ is angle between two curves tangents.

limy01+1+r42y4

limy01+1+y42y4×1+1+y4+21+1+y4+2limy01+1+y42y4(1+1+y4)+2

again factorize

limy0(1+y4)1y4(1+1+y4)+2×1+y4+11+y4+1limy01+y41y4(1+1+y4+2)(1+y4+1)limy01(1+1+y4+2)(1+y4+1)=142

ellipse x29+y21=1 and the circle x2+y2=3


Example 3: Let θ be the acute angle between the tangents to the at their point of intersection in the first quadrant. Then tanΘ is equal to:
[JEE Main 2021]
1) 523
2) 43
3) 23
4) 2

Solution

x2=3y2,x29+y2=13y2+3y2=98y2=6x2=334=94x=±32

So, the point of intersection in the first quadrant is

(32,32)
Slope of Tangent to ellipse

=m1:19x1y1=39 slope of Tangent to circle =m2:x1y1=3

tanθ=|m1m21+m1m2|=|3391+39|

=8312=233=23
Hence, the answer is the option (3).


Example 4: An angle of intersection of the curves x2a2+y2 b2=1 and x2+y2=ab,a>b is :
[JEE Main 2021]

 1) tan1(2ab)tan1(a+bab) 2) tan1(abab) 4) tan1(ab2ab)

Solution

Find a point of intersection of the curves

x2=aby2aby2a2+y2b2=1y2(1b21a2)=1aba2=1bay2(a2b2)a2b2=abay2=ab2a+by=baa+bx2=abab2a+b=a2ba+bx=aba+b So, (x1,y1)=(aba+b,baa+b)C1:x2a2+y2b2=1

Slope of tangent at (x1,y1)

m1=x1b2y1a2=abba×b2a2=(ba)3/2C2:x2y2=ab

Slope of tangent at (x1,y1)

m2=x1y1=abba=(ab1/2)tanθ=|m1m21+m1m2|:|bbaa+ab1+bbaa×ab|=|a2b2aaba+ba|θ=tan1(abab)

Hence, the answer is the option (2).

Example 3: x2α+y24=1 and y3=16x intersect at right angles, then a value of α is:
[JEE Main 2013]
1) 2
2) 43
3) 12
4) 34

Solution

x2α+y24=12xα+2y4dydx=0dydx=4xαyy3=16x3y2dydx=16dydx=163y2

Since, the curves intersect at right angles, then

4xαy×163y2=13αy3=64xα=64x3×16x=43

Hence, the answer is the option (2).

Frequently Asked Questions (FAQs)

1. What is the angle of intersection?

 The angle of intersection of two curves is defined as the angle between the tangents to the two curves at their point of intersection

2. What are the types of angles?

The three types of angles are acute, obtuse, and orthogonal.

3. When do two curves touch each other?

Two curves touch each other if the tangents to each of them are parallel to each other.

4. What are orthogonal curves?

 If the angle of the intersection of two curves is a right angle then two curves are called orthogonal curves.

5. What is a condition of orthogonal?

 Condition of Orthogonality in parametric form Where x=f(t),y=f(t) then fxgx+fygy=0

6. What is the angle of intersection between two curves?
The angle of intersection between two curves is the angle formed by the tangent lines of the two curves at their point of intersection. It represents the smallest angle between these tangent lines and provides a measure of how the curves cross each other.
7. How is the angle of intersection calculated?
The angle of intersection is calculated using the formula: tan θ = |m1 - m2| / (1 + m1m2), where θ is the angle of intersection, m1 is the slope of the tangent line to the first curve at the point of intersection, and m2 is the slope of the tangent line to the second curve at the same point.
8. Why do we use the absolute value in the formula for the angle of intersection?
We use the absolute value |m1 - m2| in the numerator to ensure that we always get a positive angle. The angle of intersection is defined as the smallest angle between the tangent lines, which is always acute (between 0° and 90°) or right (90°).
9. What does it mean if the angle of intersection is 0°?
If the angle of intersection is 0°, it means that the two curves are tangent to each other at the point of intersection. In other words, they touch at a single point but do not cross each other.
10. Can the angle of intersection be greater than 90°?
No, the angle of intersection is always defined as the smallest angle between the tangent lines, which is always between 0° and 90° (inclusive). Even if the curves appear to intersect at a larger angle, we consider the complementary acute angle.
11. How does the concept of angle of intersection extend to surfaces in 3D space?
For surfaces in 3D space, we consider the angle between the normal vectors of the surfaces at their intersection. This is analogous to considering the angle between tangent lines for curves in 2D.
12. How does the concept of angle of intersection extend to higher-dimensional spaces?
In higher-dimensional spaces, the angle between subspaces (which could be curves, surfaces, or higher-dimensional objects) is defined using the concept of principal angles. The calculation becomes more complex but the fundamental idea of measuring the "difference in direction" remains the same.
13. How do you find the slopes (m1 and m2) needed for the angle of intersection formula?
To find the slopes, you need to differentiate the equations of both curves with respect to x. Then, evaluate these derivatives at the point of intersection to get m1 and m2.
14. What happens if one of the curves is vertical at the point of intersection?
If one curve is vertical at the point of intersection, its slope is undefined (infinity). In this case, the angle of intersection is 90° (perpendicular) to the other curve's tangent line.
15. How does the angle of intersection relate to the concept of orthogonality?
Two curves are orthogonal if they intersect at a right angle (90°). So, if the angle of intersection is 90°, the curves are orthogonal at that point.
16. Can two curves intersect at multiple points with different angles?
Yes, two curves can intersect at multiple points, and the angle of intersection may be different at each point. You would need to calculate the angle separately for each intersection point.
17. What's the significance of the denominator (1 + m1m2) in the angle of intersection formula?
The denominator (1 + m1m2) in the formula accounts for the relative orientation of the two tangent lines. It ensures that the formula works correctly for all cases, including when the slopes have the same or opposite signs.
18. How does the angle of intersection formula relate to the dot product of vectors?
The angle of intersection formula is derived from the dot product formula for the angle between two vectors. In this case, the vectors are the direction vectors of the tangent lines to the curves at the intersection point.
19. What's the difference between the angle of intersection and the angle between curves?
There is no difference. The terms "angle of intersection" and "angle between curves" are often used interchangeably to describe the angle formed by the tangent lines of two curves at their point of intersection.
20. How do you determine which angle to use when two curves intersect?
Always use the smaller of the two angles formed by the intersecting curves. This is why the formula gives an angle between 0° and 90°. If you visualize a larger angle, simply use its complement (90° - θ).
21. Can the angle of intersection be used to determine if two curves are perpendicular?
Yes, if the angle of intersection is 90° (or π/2 radians), then the two curves are perpendicular at that point of intersection.
22. What does it mean if the angle of intersection is 45°?
An angle of intersection of 45° means that the two curves cross each other halfway between being parallel (0°) and perpendicular (90°). It indicates that the tangent lines to the curves at the intersection point form an isosceles right triangle.
23. How does the concept of angle of intersection apply to 3D curves?
In 3D, the angle of intersection between two curves is still defined as the angle between their tangent lines at the point of intersection. However, the calculation becomes more complex, involving the dot product of 3D vectors.
24. Can the angle of intersection be used to determine the direction of curvature?
No, the angle of intersection alone doesn't provide information about the direction of curvature. It only gives information about how the curves cross each other at a specific point.
25. How is the angle of intersection related to the concept of normal lines?
The normal line to a curve at a point is perpendicular to the tangent line at that point. If two curves intersect at 90°, the normal line of one curve will be parallel to the tangent line of the other curve at the intersection point.
26. What's the relationship between the angle of intersection and the derivative of the curves?
The angle of intersection is calculated using the slopes of the tangent lines, which are determined by the derivatives of the curves at the intersection point. Thus, the angle of intersection is directly related to the derivatives of the curves.
27. How does the angle of intersection change as you move along intersecting curves?
The angle of intersection can change as you move along intersecting curves. It depends on how the slopes of the tangent lines to each curve change. Some curves may intersect at constant angles, while others may have varying angles of intersection.
28. Can the angle of intersection be used to determine if two curves are similar?
No, the angle of intersection alone cannot determine if two curves are similar. Similarity involves more than just the angle at which curves intersect; it requires proportional dimensions and congruent angles throughout the curves.
29. How does the concept of angle of intersection apply to polar curves?
For polar curves, the angle of intersection is still defined as the angle between the tangent lines at the point of intersection. However, the calculation of slopes may involve converting from polar to Cartesian coordinates or using polar derivatives.
30. What's the significance of the angle of intersection in real-world applications?
The angle of intersection is important in various fields such as physics (for understanding collision angles), engineering (for designing intersections and junctions), and computer graphics (for rendering intersecting objects accurately).
31. How does the angle of intersection relate to the concept of tangency?
When two curves are tangent to each other, their angle of intersection is 0°. This means that the curves touch at a single point and have the same tangent line (and thus the same slope) at that point.
32. Can the angle of intersection be negative?
No, the angle of intersection is always defined as a positive value between 0° and 90°. The formula ensures this by using the absolute value of the difference in slopes.
33. How does the angle of intersection change if you reverse the direction of one of the curves?
Reversing the direction of one curve doesn't change the angle of intersection. The angle is determined by the geometric relationship between the curves at the intersection point, not by the direction in which they're traced.
34. What happens to the angle of intersection formula if the curves intersect at multiple points?
The formula remains the same, but you need to apply it separately at each intersection point. The angles of intersection may be different at each point, depending on the local behavior of the curves.
35. How is the angle of intersection related to the concept of level curves in multivariable calculus?
In multivariable calculus, the angle of intersection between level curves of two functions can provide information about the relationship between those functions. For example, orthogonal level curves indicate that the functions are independent of each other.
36. Can the angle of intersection be used to determine if two curves will intersect again?
No, the angle of intersection at one point doesn't provide information about future intersections. To determine if curves will intersect again, you need to analyze their entire equations and behaviors.
37. How does the angle of intersection relate to the concept of gradient in vector calculus?
In vector calculus, the gradient of a scalar field is perpendicular to its level curves. If two scalar fields have intersecting level curves, the angle between their gradients at the intersection point is complementary to the angle of intersection of the level curves.
38. What's the relationship between the angle of intersection and the concept of orthogonal trajectories?
Orthogonal trajectories are curves that intersect every member of a given family of curves at right angles. In this case, the angle of intersection between any curve in the family and the orthogonal trajectory is always 90°.
39. How does the angle of intersection change if you apply a linear transformation to both curves?
Linear transformations that preserve angles (like rotations and uniform scaling) will not change the angle of intersection. However, transformations that don't preserve angles (like non-uniform scaling or shearing) can change the angle of intersection.
40. Can the angle of intersection be used to determine the relative velocities of two objects following curved paths?
Yes, in physics, the angle of intersection between the paths of two objects can be used along with their speeds to determine their relative velocity vector at the point of intersection.
41. What's the relationship between the angle of intersection and the concept of directional derivatives?
The angle of intersection is related to directional derivatives in that it provides information about how rapidly one curve is changing relative to the other at the intersection point. The larger the angle, the more the curves are changing in different directions.
42. How does the angle of intersection relate to the concept of curvature?
While the angle of intersection doesn't directly measure curvature, it can provide insight into how sharply curves are bending relative to each other at the intersection point. However, curvature is a more local property of a single curve.
43. Can the angle of intersection be used to determine if two curves will eventually intersect if extended?
No, the angle of intersection only provides information about existing intersection points. To determine if curves will eventually intersect, you need to analyze their equations and asymptotic behavior.
44. How does the angle of intersection change under a conformal mapping?
Conformal mappings preserve angles. Therefore, if you apply a conformal mapping to two intersecting curves, the angle of intersection will remain the same in the transformed space.
45. What's the significance of the angle of intersection in complex analysis?
In complex analysis, the angle of intersection between curves can provide information about the behavior of complex functions. For example, analytic functions preserve angles, which is a key property in conformal mapping.
46. How does the angle of intersection relate to the concept of divergence in vector fields?
While not directly related, both concepts involve the behavior of curves or vector fields at specific points. The divergence measures the "spreading out" of a vector field, which can affect how curves defined by that field might intersect.
47. Can the angle of intersection be used to determine the stability of a system in dynamical systems theory?
While the angle of intersection alone isn't typically used to determine stability, it can provide insights into the behavior of trajectories in phase space. For example, trajectories intersecting at right angles might indicate certain types of bifurcations.
48. How does the angle of intersection relate to the concept of geodesics in differential geometry?
In differential geometry, geodesics are curves that locally minimize distance. The angle of intersection between a geodesic and another curve on a surface can provide information about the curvature of the surface and the nature of the non-geodesic curve.
49. What's the relationship between the angle of intersection and the concept of characteristic curves in partial differential equations?
Characteristic curves in PDEs are curves along which information propagates. The angle of intersection between characteristic curves and level curves of the solution can provide information about how the solution varies in different directions.
50. How does the angle of intersection relate to the concept of phase in oscillatory systems?
In oscillatory systems, the angle of intersection between trajectories in phase space can provide information about the phase relationship between different components of the system.
51. Can the angle of intersection be used to analyze the behavior of solutions to differential equations?
Yes, the angle of intersection between solution curves of differential equations can provide insights into the behavior of the system. For example, perpendicular intersections might indicate certain types of critical points or separatrices.
52. How does the concept of angle of intersection apply to fractal geometry?
In fractal geometry, the angle of intersection between different parts of a fractal can be a key characteristic of the fractal's structure. Some fractals maintain consistent intersection angles at different scales, contributing to their self-similarity.
53. What's the significance of the angle of intersection in computer vision and image processing?
In computer vision, the angle of intersection between edges or contours in an image can be used for feature detection, object recognition, and understanding the geometry of scenes.
54. How does the angle of intersection relate to the concept of cross-product in vector algebra?
The magnitude of the cross-product of two vectors is related to the sine of the angle between them. This relationship is analogous to how the angle of intersection is calculated using the tangent of the angle between curve tangents.
55. Can the angle of intersection be used to analyze the behavior of fluid flow?
Yes, in fluid dynamics, the angle of intersection between streamlines can provide information about the flow's behavior, such as areas of convergence, divergence, or rotation.

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