study material-mathematics-differential calculus
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 = bex/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 = bex/a ............................................... (i)
differentiating curve y = bex/a w.r.t. x.
& Let P(x1, y1) lie on curve
Length of subnormal =
So, subnormal varies as square of ordinate.