Orthogonal Trajectory Of Family Of Curves :
Orthogonal trajectory of family of curves is defined as a curve which cuts each members of family of curve at right angle, it is not necessary for the orthogonal trajectory curve to cross every family member of curve but wherever it crosses the family members angle must be
between them, let’s learn about the keyword’s trajectory and orthogonal.
Trajectory:
trajectory is known as a path formed by the moving point for example if a ball being thrown upward than its path or trajectory would by parabolic in downward shape, here ball is behaving as a moving point.
Orthogonal:
If angle between tangents drawn on two curves at there point of intersection is
than it is known as orthogonal intersection orthogonal is a Greek word originated from orthogonios, it has two meanings in geometry it represents two lines meeting at right angle and in probability two events which are independent of each other.
Orthogonal Trajectory Of Family Of Curves Introduction
Orthogonal Trajectory Of Family Of Curves Introduction:
To find orthogonal trajectory of family of curves there is well defined algorithm which must be followed let’s discuss it here.
Algorithm
(1)
Represent family of curves as
f
(
x.y.c
) = 0 here c is an arbitrary constant
(2)
Differentiate
f
(
x.y.c.
) = 0 with respect to independent variable and form the differential equation which does not contain arbitrary constant.
(3)
Now replace
with
in the differential equation obtained.
(4)
Integrate obtained differential equation by using any suitable method. Final Cartesian equation will represent the orthogonal trajectories of the given family of curves, let’s explore the algorithm by examples.
Orthogonal Trajectory Of Family Of Curves Examples
Example 1:
Find orthogonal trajectory for family of curves represented as
y
=
ax
2
, (a is an arbitrary constant)
Solution:
We have equation of family of curves as
y
=
ax
2
, let’s differentiate it with respect to
x
.
{
}
Replace
with
Integrate both sides
{denoted family of curves which are orthogonal to family members of
y
=
ax
2
}
Example 2:
Find orthogonal trajectory for family of curves represented as
y
=
mx
? (m is an arbitrary constant)
Solution:
Family of curves is represented as
y
=
mx
, let’s differentiate it with respect to
x
.
{by
y
=
mx
}
Replace
with
Integrate both sides.
{
R
= –2
c
}{denoted family of curves which are orthogonal to family members of
y
=
mx
}
Above equation represents family of circles centre at origin and radius as
.
Example 3:
Find orthogonal trajectories for the circles
, (b is an arbitrary constant.)
Solution:
Lets differentiate family of curves equation with respect to
x
.
{by
}
Replace
with
Integrate both sides
{denoted family of curves which are orthogonal to family members of
}
Above equation also represents a circle, since family of curves and orthogonal trajectory both are same hence it is known as self-orthogonal.
Orthogonal Trajectory Of Family Of Curves Rapid Question
(1)
Find orthogonal trajectory for
where
b
is an arbitrary constant.
(2)
Find orthogonal trajectory for
where
p
is an arbitrary content.
Orthogonal Trajectory Of Family Of Curves Illustration
(1)
Find orthogonal trajectory for family of curve represented as
where
b
is an arbitrary constant.
Solution:
Family of curve is given as
. Let’s differentiate it with respect to
x
.
{
}
{
}
Let’s replace
with
Integrate both sides
{denoted family of curves which are orthogonal to family members of
}
(2)
Find orthogonal trajectory for family of curve represented as
where a is an arbitrary constant.
Solution:
Family of curve is given as
let’s differentiate it with respect to
x
.
Let’s replace
with
Integrate both sides
{denoted family of curves which are orthogonal to family members of
}
Above equation represents family of downward parabola.
(3)
Find orthogonal trajectory for
where m is an arbitrary constant.
Solution:
Family of curve is given as
let’s differentiate it with respect to
x
.
Replace
with
Integrate both sides
{denoted family of curves which are orthogonal to family members of
}
Orthogonal Trajectory Of Family Of Curves Rapid Question
(1)
Find orthogonal trajectory for y = a + log b x where
b
is an arbitrary constant.
(2)
Find orthogonal trajectory for
, where
m
is arbitrary constant.
