Length of Tangent , Normal , Sub-tangent and Sub-normal

Let P (x, y) be any point on y = f (x). Let the tangent drawn at ‘ P ‘ meets the x-axis at ‘ T ‘, and normal drawn at ‘ P ‘ meets the x-axis at ‘ N ‘.

PT is called length of the tangent and PN is called the length of the normal.

If ‘ P1 ‘ be the projection of the point P on the x-axis then TP1 is called the sub-tangent (projection of line segment PT on the x-axis) and NP1 is called the sub normal (projection of line segment PN on the x-axis).

Let ∠PTN = θ => ∠P1PN = θ

We have tanθ = dy/dx and PP1 = |y|

Now, PT= |y cosec θ|
or, PT =

Hence length of the tangent PT =

Now, PN = |y sec θ| =

Length of the normal, PN =

Now, TP1 = |y cot θ| =

=> Sub-tangent TP1 =

Finally, NP1 = |y tan θ| =

=> Sub-normal, NP1 =

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