10. Problem: Form the differential equation corresponding to the family of circles passing
through the origin and having centres on Y-axis.
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hii here is ur answer!!
ANSWER
It is given that, circles pass through origin and their centres lie on Y-axis.
Let (0,k) be the centre of the circle with k as its centre.
So, the equation of circle is
(x−0)
2
+(y−k)
2
=k
2
x
2
+(y−k)
2
=k
2
x
2
+y
2
−2ky=0
2y
x
2
+y
2
=k .....(1)
Differentiating equation 1 w.r.t. x, we get,
4y
2
2y(2x+2y
dx
y
)−(x
2
+y
2
)
dx
2dy
=0
4y(x+y
dx
dy
)−2(x
2
+y
2
)
dx
dy
=0
4xy=4y
2
dx
dy
−2(x
2
+y
2
)
dx
dy
=0
(4y
2
−2x
2
−2y
2
)
dx
dy
+4xy=0
(2y
2
−2x
2
)
dx
dy
+4xy=0
(y
2
−x
2
)
dx
dy
+2xy=0
(x
2
−y
2
)
dx
dy
−2xy=0
hope helpful!!
:)
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