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Length of Tangent and Normal and Subtangent and subnormal - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Length of Tangent, Normal, Subtangent and subnormal is considered one the most difficult concept.

  • 15 Questions around this concept.

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QuestionMEDIUM

If   2a+3b+6c= 0,   then at least one root of the  equation

ax^{2}+bx+c= 0, lies in the interval

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Length of Tangent, Normal, Subtangent and subnormal

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(xo, yo)

In the figure, the length of segment PT is the length of the tangent.

In ΔPTS

\\\text{PT}=|y\cdot \csc\theta|=|y|\sqrt{1+\cot^2\theta}\\\\\mathrm{\;\;\;\;\;\;=|y|\sqrt{1+\left ( \frac{dx}{dy} \right )_{\left (x_0,y_0 \right )}}}

 

Length of Normal:

A segment of normal PN  is called length of Normal.

In ΔPSN

\\\text{PN}=|y\cdot \csc\left (90^{\circ}-\theta \right )|=|y\cdot \sec\theta|\\\\\mathrm{\;\;\;\;\;\;=|y|\sqrt{1+\tan^2\theta}=|y|\sqrt{1+\left ( \frac{dy}{dx} \right )_{\left (x_0,y_0 \right )}}}

 

Length of Subtangent:

The projection of the segment PT along the x-axis is called the length of subtangent. In the figure, ST is the length of subtangent. 

In ΔPST

\\\text{ST}=|y\cdot \cot\theta|=\left | \frac{y}{\tan\theta} \right |\\\\\mathrm{\;\;\;\;\;=\left | y\cdot\frac{dx}{dy} \right |}

 

Length of Subnormal:

The projection of the segment PN along the x-axis is called the length of subnormal. In the figure, SN is length of subnormal. 

In ΔPSN

\\\text{SN}=|y\cdot \cot\left (90^{\circ}-\theta \right )|=|y\cdot\tan\theta|\\\\\mathrm{\;\;\;\;\;=\left | y\cdot\frac{dy}{dx} \right |}

Study other Related Concepts

Length of Tangent and Normal and Subtangent and subnormal Current Topic

Limit
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

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