Length of Tangent, Normal, Subtangent, and Subnormal is an important concept in calculus. It is useful in understanding the relationship between curves and their slopes. 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 Tangents and slopes have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.
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In this article, we will cover the concept of the Length of Tangent, Normal, Subtangent, and Subnormal. 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 four questions have been asked on this topic in JEE Main from 2013 to 2023, including one question in 2014, one question in 2015, one question in 2019, and one in 2021.
Tangent
The tangent to a curve at a point
The slope of the tangent to the curve
NORMAL
The normal to the curve at any point P on it is the straight line which passes through P and is perpendicular to the tangent to the curve at P
Length of Tangent:
The length of the portion lying between the point of tangency i.e. the point on the curve from which a tangent is drawn and the point where the tangent meets the
In the figure, the length of segment PT is the length of the tangent.
In
Length of Normal:
A segment of normal PN is called length of Normal.
In
Length of Subtangent:
The projection of the segment PT along the x-axis is called the length of the subtangent. In the figure, ST is the length of the subtangent.
In ΔPST
Length of Subnormal:
Example 1: If the Rolle's theorem holds for the function
[JEE Main 2014]
1) -1
2) 1
3) 2
4) -2
Solution
As we have learned
Rolle's Theorems -
Let
1.
2.
3.
Geometrical interpretation of Rolle's theorem -
Let f(x) be a function defined on [a, b] such that the curve y = f(x) is continuous between points {a, f(a)} and {b, f(b)} at every points on the curve encept at the end point it is possible to draw a unique tangent and ordinates at x = a and x = b are equal f(a) = f(b).
Example 2: If the tangent to the curve
[JEE Main 2021]
1)
2)
3) 0
4)
Solution
Equation of tangent at
now solve the above equation with
Hence, the answer is the option (1).
Example 3: The shortest distance between the line
[JEE Main 2015]
1)
2)
3)
4)
Solution
Line
Eq. of tangent to
Tangent at
so, slope should be equal
put the value of
so,
Perpendicular distance from the point
Hence, the answer is option (1).
Example 4: If Rolle's theorem holds for the function
[JEE Main 2019]
1) -1
2) -2
3) -3
4) -4
Solution
Rolle's Theorems
Let
1.
2.
3.
Then there is at least one c lying in
Now,
It is continuous and differentiable in any interval as it is a polynomial, hence it is continuous in
Now
where,
Also
As Rolle's Theorem is satisfied at
So,
Hence the answer is the option (1)
Example 5 : Length of normal drawn to the curve
1)
2)
3)
4)
Solution
As we know, the length of Normal
Here the curve is
A tangent is a straight line that touches a curve at a single point without crossing it at that point.
The equation of the tangent at
The length of the portion lying between the point of tangency i.e. the point on the curve from which a tangent is drawn and the point where the tangent meets the
The projection of the segment PT along the
The projection of the segment PN along the x -axis is called the length of the subnormal.
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