Question:
from the differntial equation of the family of the circles having centre on y-axis and radius 3units
Answer:
Given radius of circle = 3
Now equation of the circle having centre on y-axis and radius 3 is
(x-0)2 + (y-h)2 = 32
=> x2 + (y-h)2 = 9 ..............1
Now differentiate with respect to x
2x + 2(y-h)* dy/dx = 0
=> x + (y-h)* dy/dx = 0
=> y-h = -x/(dy/dx) ............2
Now put this value in equation 1, we get
x2 + {-x/(dy/dx)}2 = 9
=> x2 + x2 /(dy/dx)2 = 9
=> x2 *(dy/dx)2 + x2 = 9*(dy/dx)2
=> x2 *(dy/dx)2 + x2 - 9*(dy/dx)2 = 0
=>(x2 - 9)*(dy/dx)2 + x2 = 0
So differntial equation of the family of the circles having centre on y-axis and radius 3units is (x2 - 9)*(dy/dx)2 + x2 = 0
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