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.
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.
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
Let
Let
Then
From the figure it follows,
The intersection of these curves is defined as the acute angle between the tangents.
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
Let
from the figure
If the angle of the intersection of two curves is a right angle then two curves are called orthogonal curves.
In this case,
this is also the condition for two curves to be orthogonal.
Condition for two curves to touch each other
Example 1: If the curves
[JEE Main 2028]
1)
2) 6
3)
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:
Slope of tangent of first curve
Slope of tangent of second curve
So
Example 2:
[JEE Main 2019]
1) exists and equals
2) exists and equals
3) exists and equals
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
where
again factorize
ellipse
Example 3: Let
[JEE Main 2021]
1)
2)
3)
4) 2
Solution
So, the point of intersection in the first quadrant is
Slope of Tangent to ellipse
Hence, the answer is the option (3).
Example 4: An angle of intersection of the curves
[JEE Main 2021]
Solution
Find a point of intersection of the curves
Slope of tangent at
Slope of tangent at
Hence, the answer is the option (2).
Example 3:
[JEE Main 2013]
1) 2
2)
3)
4)
Solution
Since, the curves intersect at right angles, then
Hence, the answer is the option (2).
The angle of intersection of two curves is defined as the angle between the tangents to the two curves at their point of intersection
The three types of angles are acute, obtuse, and orthogonal.
Two curves touch each other if the tangents to each of them are parallel to each other.
If the angle of the intersection of two curves is a right angle then two curves are called orthogonal curves.
Condition of Orthogonality in parametric form Where
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