find the orthogonal traiectory of the
curve porabola y²=4a(x+a)and it is also
self orthoganal
Answers
Step-by-step explanation:
Answer
Given the equation of the family of parabolas is y
2
=4ax
Here the parameter is a, which is also an arbitrary constant for finding the ordinary differential equation.
Now differentiating the equation with respectto x on both sides gives,
dx
dy
=
y
2a
∴a=
2
y
(
dx
dy
)
substituting in the equation of the family of curves gives,
y
2
=2xy(
dx
dy
) which is differential equation of the family of parabolas.
Now,to find the equation of the orthogonal trajectories we need to replace (
dx
dy
) by (
dy
−dx
) and we need to solve it back
y
2
=2xy(
dy
−dx
)
Regrouping the terms and integrating gives,
∫ydy=∫(−2x)dx
⟹
2
y
2
=−x
2
+c where c is the integration constant
regrouping the terms gives,
2x
2
+y
2
=C
2
where C is a constant
this is the modal