Doubt from differentiation equation

Question 1 second and third part and secondly i want to know for first part i am getting (x^2-y^2)dy/dx=2xy but i dont know why the answer key has double differentiated it

Part 2)
y=(dy/dx)x
Is it correct?

Yes

Thanks

Can you tell fourth part too if possible

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In the first part , you will get 2 arbitrary constants;
one for the radius and the other for the coordinates of the centre of the circle.
So it is differentiated twice.

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Question 9 i solved by taking y-3/x+1=t but always stuck at one point help in this


After this you can directly integrate and get the answer.

I understood my mistake i got the answer thanks for your working