application-of-derivatives-2

Dumb Question: How this derived ?

Ans: Let   m₁ = tan ₁ =
              m₂ = tan ₂ =
From fiq,   = + ₁
           = ₂ - ₁
          
          


Orthogonal curves: If angle of intersection of two curves is right angle, two curves are c/d orthogonal curves.
If curves are orthogonal,  =



Illustration: Find angle of intersection of curves   y = x² & y = 4 - x²
For intersection points of given curves,
(x²) = 2x
(4 - x²) = - 2x
At  x = - ,
= 2 x = 2   &   = - 2



  
Both angles are equal.


Length of Tangent, Sub-Tangent, Normal & sub-Normal:

Length of Tangent: Length of segment PT of tangent b/w point of tangent & x-axis is c/d length of tangent.

PT =

Subtangent: Projection of segment PT along x-axis i.e. St c/d subtangent.

Length of Normal: Length of segment PN intercapted b/w point on curve r x-axis.
PN =

Subnormal: Projection of segment PN along x-axis i.e. c/d subnormal.
SN = |y,


Dumb Question: How these relation derived ?

Ans: Since PT makes angle with x-axis, then
tan =

Subtangent = ST = PS cot
But PS = y₁   &   cot = [see in fig.]

= cot(90 - )  PS tan
Subnormal = SN =
Length of tangent = PT =
Length normal = PN =


Illustration: Show that curve y = be^x/a. subnormal varies as square of ordinate ?

Ans:
[Dumb Question: What is ordinate ?

Ans: y-axis component is c/d ordi i.e. in P(x₁, y₁)y₁ is ordinate of point P.
y = be^x/a ............................................... (i)
differentiating curve   y = be^x/a   w.r.t. x.

& Let P(x₁, y₁) lie on curve

Length of subnormal =
                             
So, subnormal varies as square of ordinate.