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 =