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Length of Tangent, Subtangent, Normal and Subnormal

Length of Tangent, Subtangent, Normal and Subnormal

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

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.

Length of Tangent, Subtangent, Normal and Subnormal
Length of Tangent, Subtangent, Normal and Subnormal

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 to the Curve at point:

Tangent

The tangent to a curve at a point $P$ on it is defined as the limiting position of the secant $P Q$ as the point $Q$ approaches the point $P$ provided such a limiting position exists.
The slope of the tangent to the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ at the point $\left(\mathrm{x}_0, \mathrm{y}_0\right)$ is given by $\left.\frac{d y}{d x}\right]_{\left(x_0, y_0\right)} \quad\left(=\mathrm{f}^{\prime}\left(\mathrm{x}_0\right)\right)$. So the equation of the tangent at $\left(\mathrm{x}_0, \mathrm{y}_0\right)$ to the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ is given by $y-y_0=f^{\prime}\left(x_0\right)\left(x-x_0\right)$.

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, Normal, Subtangent and Subnormal

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 $x$-axis. Here point of tangency is $P\left(x_0, y_0\right)$
In the figure, the length of segment PT is the length of the tangent.
In $\triangle \mathrm{PTS}$

$
\begin{aligned}
\mathrm{PT} & =|y \cdot \csc \theta|=|y| \sqrt{1+\cot ^2 \theta} \\
& =|\mathrm{y}| \sqrt{1+\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)_{\left(\mathrm{x} 0, \mathrm{y}_0\right)}}
\end{aligned}
$

Length of Normal:

A segment of normal PN is called length of Normal.
In $\triangle P S N$

$
\begin{aligned}
\mathrm{PN} & =\left|y \cdot \csc \left(90^{\circ}-\theta\right)\right|=|y \cdot \sec \theta| \\
& =|\mathrm{y}| \sqrt{1+\tan ^2 \theta}=|\mathrm{y}| \sqrt{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(\mathrm{x} 0, \mathrm{y} 0)}}
\end{aligned}
$

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

$\begin{aligned} \mathrm{ST} & =|y \cdot \cot \theta|=\left|\frac{y}{\tan \theta}\right| \\ & =\left|\mathrm{y} \cdot \frac{\mathrm{dx}}{\mathrm{dy}}\right|\end{aligned}$

Length of Subnormal:

$
\begin{aligned}
&\text { The projection of the segment PN along the } \mathrm{x} \text {-axis is called the length of the subnormal. In the figure, } \mathrm{SN} \text { is the length of }\\
&\begin{aligned}
& \operatorname{In} \triangle \mathrm{PSN} \\
& \mathrm{SN}=\left|y \cdot \cot \left(90^{\circ}-\theta\right)\right|=|y \cdot \tan \theta| \\
&=\left|\mathrm{y} \cdot \frac{\mathrm{dy}}{\mathrm{dv}}\right|
\end{aligned}
\end{aligned}
$

Recommended Video Based on Length of Tangents, Normal, Subtangent and Subnormal


Solved Examples Based On Length of tangents, normal, subtangent, and subnormal:

Example 1: If the Rolle's theorem holds for the function $f(x)=2 x^3+a x^2+b x$ in the interval $[-1,1]$ for the point $c=\frac{1}{2}$ then the value of $2 a+b$ is :
[JEE Main 2014]
1) -1
2) 1
3) 2
4) -2

Solution
As we have learned
Rolle's Theorems - $\square$
Let $f(x)$ be a function of $x$ subject to the following conditions.
1. $\mathrm{f}(\mathrm{x})$ is continuous function of $x: x \in[a, b]$
2. $\mathrm{F}(\mathrm{x})$ is exists for every point: $x \epsilon[a, b]$
3. $f(a)=f(b)$ then $f^{\prime}(c)=0$ such that $a<c<b$.

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).

$
\begin{aligned}
&\text { - wherein, We have }\\
&\begin{aligned}
& f^{\prime}(1 / 2)=\frac{f(1)-f(-1)}{2}=0 \\
\Rightarrow & \left.\left(6 x^2+2 a x+b\right)\right|_{x=1 / 2} \\
& \frac{2+a+b-(-2+a-b)}{2}=0 \\
\Rightarrow & 3 / 2+a+b=\frac{4+2 b}{2}=0 \\
\Rightarrow & 3+2 a+2 b=4+2 b=0 \\
\Rightarrow & a=1 / 2 \text { and } b=-2 \\
\therefore & 2 a+b=-1
\end{aligned}
\end{aligned}
$

Example 2: If the tangent to the curve $y=x^3$ at the point $P\left(t, t^3\right)$ meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio $1: 2$ is:
[JEE Main 2021]
1) $-2 t^3$
2) $-t^3$
3) 0
4) $2 t^3$

Solution
Equation of tangent at $\mathrm{P}\left(\mathrm{t}, \mathrm{t}^3\right)$

$
\left(y-t^3\right)=3 t^2(x-t)
$

now solve the above equation with

$
y=x^3
$

$
\begin{aligned}
&\begin{aligned}
& \text { By }(1) \&(2) \\
& x^3-t^3=3 t^2(x-t) \\
& \mathrm{x}^2+\mathrm{xt}+\mathrm{t}^2=3 \mathrm{t}^2 \\
& \mathrm{x}^2+\mathrm{xt}-2 \mathrm{t}^2=0 \\
& (x-t)(x+2 t)=0 \\
& \Rightarrow x=-2 t \Rightarrow Q\left(-2 t,-8 t^3\right)
\end{aligned}\\
&\text { Ordinate of required point }\\
&=\frac{2 t^3+\left(-8 t^3\right)}{3}=-2 t^3
\end{aligned}
$

Hence, the answer is the option (1).

Example 3: The shortest distance between the line $\mathrm{y}=\mathrm{x}$ and the curve $y^2=x-2$ is.
[JEE Main 2015]
1) $\frac{7}{4 \sqrt{2}}$
2) $\frac{7}{2 \sqrt{2}}$
3) $\frac{7}{4 \sqrt{3}}$
4) $\frac{5}{4 \sqrt{2}}$

Solution

Line $\mathrm{y}=\mathrm{x}$
Eq. of tangent to $\mathrm{y}^2=\mathrm{x}-2$
$y^2=x-2$
$2 \mathrm{yy}^{\prime}=1$
$\mathrm{y}^{\prime}=\frac{1}{2 \mathrm{y}}=$ slope
Tangent at $P$ is parallel to the line $x=y$
so, slope should be equal

$
\mathrm{y}^{\prime}=1=\frac{1}{2 \mathrm{y}} \Rightarrow \mathrm{y}=\frac{1}{2}
$

put the value of $y$ in the curve $y^2=x-2$

$
\left(\frac{1}{2}\right)^2=x-2 \Rightarrow x=\frac{9}{4}
$

so, $\mathrm{P}=\left(\frac{9}{4}, \frac{1}{2}\right)$
Perpendicular distance from the point $P$ to the line $y=x$

$
\left|\frac{\left(\frac{9}{4}-\frac{1}{2}\right)}{\sqrt{1^2+1^2}}\right|=\frac{7}{4 \sqrt{2}}
$
Hence, the answer is option (1).

Example 4: If Rolle's theorem holds for the function $f(x)=2 x^3+b x^2+c x, x \in[-1,1]$ at the point $x=\frac{1}{2}$, then $2 b+c$ equals:
[JEE Main 2019]
1) -1
2) -2
3) -3
4) -4

Solution
Rolle's Theorems
Let $f(x)$ be a function with the following properties
1. $f(x)$ is a continuous function in $[a, b]$
2. $\underline{\underline{f}}^{\prime}(\mathrm{x})$ exists for every point in (a,b)
3. $f(a)=f(b)$

Then there is at least one c lying in $(a, b)$ such that $\underline{f}(c)=0$
Now,

$
f(x)=2 x^3+b x^2+c x
$

It is continuous and differentiable in any interval as it is a polynomial, hence it is continuous in $[-1,1]$ and differentiable in $(-1,1)$
Now $f(-1)=f(1)$
where, $f(1)=2+b+c$ and $f(-1)=-2+b-c$

$
\begin{aligned}
& \Rightarrow b+c+2=b-2-c \\
& \Rightarrow c+2=0 \\
& \therefore c=-2
\end{aligned}
$
Also

$
f^{\prime}(x)=6 x^2+2 b x+c
$
As Rolle's Theorem is satisfied at $x=1 / 2$, hence $f^{\prime}(1 / 2)=0$

$
\begin{aligned}
& 0=6\left(\frac{1}{2}\right)^2+b \times 2 \times \frac{1}{2}-2 \\
& 0=\frac{6}{4}+b-2 \\
& \mathrm{~b}=1 / 2
\end{aligned}
$
So, $2 b+c=-1$

Hence the answer is the option (1)

Example 5 : Length of normal drawn to the curve $x y=16$ at its point $(4,4)$ equals?
1) $6 \sqrt{2}$
2) $5 \sqrt{2}$
3) $4 \sqrt{2}$
4) $3 \sqrt{2}$

Solution
As we know, the length of Normal $=y_o \sqrt{1+\left(y^{\prime}\right)^2}$
Here the curve is

$
\begin{aligned}
& x y=16 \\
& \Rightarrow y=\frac{16}{x} \\
& \Rightarrow y^{\prime}=-16 / x^2 \\
& \Rightarrow y^{\prime} \text { at }(4,4)=-1 \\
& \therefore \text { Length of normal }=4 \sqrt{1+1}=4 \sqrt{2}
\end{aligned}
$

Frequently Asked Questions (FAQs)

1. What is tangent?

A tangent is a straight line that touches a curve at a single point without crossing it at that point.

2. What is the equation of tangent?

 The equation of the tangent at $\left(\mathrm{x}_0, \mathrm{y}_0\right)$ to the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ is given by $y-y_0=f^{\prime}\left(x_0\right)\left(x-x_0\right)$.

3. What is the length of the 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 $x$-axis.

4. What is the length of the subtangent?

The projection of the segment PT along the $x$-axis is called the length of the subtangent.

5. What is subnormal?

The projection of the segment PN along the x -axis is called the length of the subnormal.

6. What is a tangent line to a curve?
A tangent line is a straight line that touches a curve at a single point without crossing through it. It represents the instantaneous rate of change of the function at that point.
7. How is the length of a tangent related to the derivative of a function?
The length of a tangent is directly related to the derivative of a function. It can be calculated using the slope of the tangent line, which is equal to the derivative of the function at the point of tangency.
8. What is a normal line to a curve?
A normal line is a straight line that is perpendicular to the tangent line at the point of tangency. It intersects the curve at a right angle.
9. How are the slopes of tangent and normal lines related?
The slopes of tangent and normal lines are perpendicular to each other. If the slope of the tangent line is m, the slope of the normal line is -1/m (assuming m ≠ 0).
10. What is the subtangent?
The subtangent is the length of the line segment from the x-intercept of the tangent line to the point where a perpendicular line from the point of tangency meets the x-axis.
11. What is the difference between a tangent and a secant line?
A tangent line touches the curve at a single point, while a secant line intersects the curve at two or more points. As the secant line's points of intersection get closer together, it approaches the tangent line.
12. How do you find the equation of a tangent line?
To find the equation of a tangent line: 1) Calculate the derivative of the function. 2) Evaluate the derivative at the point of tangency to find the slope. 3) Use the point-slope form of a line equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point of tangency.
13. How do you determine if a point on a curve has a horizontal tangent?
A point on a curve has a horizontal tangent when the derivative of the function at that point is zero. This occurs at local maximum or minimum points of the function.
14. How does the concept of tangent and normal lines extend to three-dimensional surfaces?
In three dimensions, tangent lines become tangent planes, and normal lines become normal vectors. These concepts are used to describe the local behavior of surfaces and are crucial in multivariable calculus and 3D modeling.
15. How are tangent lines used in approximating functions?
Tangent lines are used to create linear approximations of functions near a specific point. This is the basis of linearization and Taylor series approximations, which are crucial in many areas of applied mathematics and physics.
16. Can a curve have a tangent line at a point of discontinuity?
Generally, a curve does not have a tangent line at a point of discontinuity. However, in some cases, one-sided tangents may exist if the function has one-sided limits at the point of discontinuity.
17. How do you find points on a curve where the tangent line is parallel to a given line?
To find such points, set the derivative of the function equal to the slope of the given line and solve for x. This gives you the x-coordinates of points where the tangent line has the same slope as the given line.
18. How do inflection points affect tangent lines?
At an inflection point, the tangent line crosses the curve, changing from being above the curve to below it (or vice versa). The second derivative of the function is zero at an inflection point.
19. How are tangent lines used in Newton's method for finding roots?
Newton's method uses tangent lines to approximate the roots of a function. It iteratively finds better approximations by following the tangent line from the current guess to its x-intercept.
20. What is the relationship between the concavity of a curve and its tangent lines?
For a concave up curve, the tangent line lies below the curve (except at the point of tangency). For a concave down curve, the tangent line lies above the curve. The concavity changes at inflection points.
21. How do you find the angle between two curves at their point of intersection?
The angle between two curves at their intersection is the same as the angle between their tangent lines at that point. Calculate the slopes of both tangent lines and use the arctangent of the difference of these slopes to find the angle.
22. How do you determine if a curve has a vertical tangent line?
A curve has a vertical tangent line at points where the derivative approaches infinity or is undefined. This often occurs at cusps or where the function's graph has a vertical asymptote.
23. What is the evolute of a curve, and how is it related to normal lines?
The evolute of a curve is the locus of all its centers of curvature. It can be thought of as the envelope of all the normal lines to the original curve. The evolute is important in the study of the geometry of curves.
24. How do tangent lines behave near a point of discontinuity in the derivative?
Near a point where the derivative is discontinuous, the tangent lines on either side of the point may have different slopes. This can result in a corner or cusp in the curve.
25. What is the significance of the point where the tangent line coincides with the curve?
If a tangent line coincides with the curve over an interval (not just at a point), it indicates that the curve is a straight line over that interval. This means the function is linear in that region.
26. What is the relationship between the tangent line and the limit definition of the derivative?
The limit definition of the derivative represents the process of secant lines approaching the tangent line as the change in x approaches zero. The resulting limit gives the slope of the tangent line.
27. What is the pedal curve, and how is it related to normal lines?
The pedal curve of a given curve with respect to a point P is the locus of the feet of perpendiculars drawn from P to the tangent lines of the original curve. It's closely related to the concept of normal lines and is used in various geometric studies.
28. How do you determine if a curve has an asymptote using concepts of tangent lines?
A curve has a horizontal asymptote if the tangent lines approach a horizontal line as x approaches infinity. It has a vertical asymptote if the tangent lines become increasingly vertical as x approaches a certain value.
29. What is the envelope of a family of curves, and how is it related to tangent lines?
The envelope of a family of curves is a curve that is tangent to each member of the family at some point. It can be thought of as the "outline" of the family of curves and is found by considering the limiting positions of intersections of nearby curves in the family.
30. How do concepts of tangent and normal lines apply to space curves?
For space curves, tangent lines become tangent vectors in three dimensions. Normal planes replace normal lines, and the concept of curvature becomes more complex, involving both the curvature and torsion of the curve.
31. How is the length of the subtangent calculated?
The length of the subtangent is calculated as |y/y'|, where y is the y-coordinate of the point of tangency and y' is the derivative of the function at that point.
32. What is the subnormal?
The subnormal is the length of the line segment from the foot of the perpendicular from the point of tangency to the x-axis, to the point where the normal line intersects the x-axis.
33. How is the length of the subnormal calculated?
The length of the subnormal is calculated as |y * y'|, where y is the y-coordinate of the point of tangency and y' is the derivative of the function at that point.
34. Why is the concept of tangent lines important in calculus?
Tangent lines are crucial in calculus because they represent the instantaneous rate of change of a function at a specific point. This concept is fundamental to understanding derivatives and their applications in various fields.
35. How does the length of a tangent change as you move along a curve?
The length of a tangent typically varies as you move along a curve. It depends on the shape of the curve and how quickly it's changing at each point. The length can increase, decrease, or remain constant depending on the function.
36. Can a curve have more than one tangent line at a point?
Generally, a smooth curve has only one tangent line at a given point. However, at cusps or corners, a curve may have multiple tangent lines or no tangent line at all.
37. What is the relationship between the tangent and the derivative at a point?
The slope of the tangent line at a point is equal to the value of the derivative of the function at that point. This relationship is fundamental to differential calculus.
38. What is the significance of the normal line in optimization problems?
The normal line is important in optimization problems because it helps identify the shortest distance between a point and a curve. This concept is used in various applications, including finding the closest point on a curve to a given point.
39. Can the subtangent or subnormal ever be zero? If so, what does this mean geometrically?
The subtangent can be zero when the tangent line is vertical (undefined slope). The subnormal can be zero when the tangent line is horizontal (slope of zero). Geometrically, a zero subtangent means the tangent line passes through the y-axis, while a zero subnormal means the normal line is vertical.
40. What is the relationship between the lengths of the tangent, normal, subtangent, and subnormal?
These lengths are related through the properties of right triangles formed by these lines. The tangent and normal form the hypotenuses of two right triangles, with the subtangent and subnormal forming the bases of these triangles respectively.
41. How does the concept of tangent lines relate to the definition of the derivative?
The derivative is defined as the limit of the slope of secant lines as they approach the tangent line. This limit gives the instantaneous rate of change, which is represented by the slope of the tangent line.
42. What is the geometric interpretation of the subtangent and subnormal?
Geometrically, the subtangent represents the x-intercept of the tangent line relative to the point of tangency, while the subnormal represents the x-intercept of the normal line relative to the foot of the perpendicular from the point of tangency.
43. Can a curve have a normal line that is parallel to the x-axis?
Yes, a curve can have a normal line parallel to the x-axis. This occurs when the tangent line at that point is vertical, which happens when the derivative is undefined or approaches infinity.
44. What is the significance of the normal line in the context of curve fitting?
In curve fitting, the normal line is used to measure the distance between a point and a curve. Minimizing the sum of squares of these normal distances is the basis for orthogonal regression, which is used when there's uncertainty in both variables.
45. How does the concept of tangent lines extend to parametric equations?
For parametric equations, the tangent line is found using the derivatives of both x and y with respect to the parameter. The slope of the tangent is dy/dt divided by dx/dt, assuming dx/dt ≠ 0.
46. What is the relationship between the radius of curvature and the normal line?
The radius of curvature is measured along the normal line from the point on the curve to the center of the osculating circle. It represents how quickly the curve is turning at that point.
47. How do you find the points on a curve where the tangent line passes through a given point?
To find such points, set up an equation using the point-slope form of the tangent line, with the slope as the derivative and the given point. Solve this equation along with the equation of the curve to find the points of tangency.
48. How does the concept of tangent and normal lines apply to implicit functions?
For implicit functions, the slope of the tangent line is found using implicit differentiation. The normal line is still perpendicular to the tangent, but its equation is derived from the implicitly differentiated equation.
49. What is the significance of points where the normal line passes through the origin?
Points where the normal line passes through the origin are significant in various geometric and physical problems. For example, in optics, these points can represent where light rays converge after reflection from a curved surface.
50. How do you find the equation of the normal line to a curve at a given point?
To find the equation of the normal line: 1) Calculate the derivative of the function. 2) Evaluate the derivative at the given point to find the slope of the tangent line. 3) The slope of the normal line is the negative reciprocal of this. 4) Use the point-slope form with this slope and the given point to write the equation.
51. How does the behavior of tangent lines change near a point of infinite curvature?
Near a point of infinite curvature, such as a cusp, the tangent lines change direction very rapidly. The normal lines in this region will intersect at or very close to the point of infinite curvature.
52. What is the relationship between the length of the tangent and the angle it makes with the x-axis?
The length of the tangent from the point of tangency to its x-intercept is related to the angle it makes with the x-axis through trigonometry. If θ is this angle, the length of the tangent is y/sin(θ), where y is the y-coordinate of the point of tangency.
53. How does the concept of tangent lines extend to curves defined by polar equations?
For polar curves, the slope of the tangent line is found using the formula dy/dx = (dr/dθ * sin(θ) + r * cos(θ)) / (dr/dθ * cos(θ) - r * sin(θ)). This involves both the r and θ derivatives of the polar function.
54. What is the significance of the point where the tangent line is perpendicular to the radius vector in polar coordinates?
In polar coordinates, when the tangent line is perpendicular to the radius vector, it indicates a point where the rate of change of the radius with respect to the angle is equal to the radius itself. This condition often occurs at interesting geometric features of polar curves.
55. What is the osculating circle of a curve, and how is it related to the concepts of tangent and normal lines?
The osculating circle is the circle that best approximates the curve at a given point. Its center lies on the normal line, and its radius is the radius of curvature at that point. The osculating circle shares the same tangent line with the curve at the point of contact.

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