Math, asked by punatavinashpunati, 5 months ago

find the orthogonal traiectory of the
curve porabola y²=4a(x+a)and it is also
self orthoganal​

Answers

Answered by karthikasridevipal
0

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

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