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  • Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

10.    Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

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Equation of the family of circles having centre on y-axis and radius 3 units
Let suppose centre is at (0,b)
Now, equation of circle with center (0,b) an  radius = 3 units
(x-0)^2+(y-b)^2=3^2 \ \ \ \ \ \ \ \ \ \ \ -(i)\\ x^2+y^2+b^2-2yb = 9 
Now, differentiate w.r.t x 
we get,
2x+2yy^{'}-2by^{'}= 0\\ 2x+2y(y-b)= 0\\ (y-b)=\frac{-x}{y^{'}} \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)
Put value fro equation (ii) in (i)
(x-0)^2+(\frac{-x}{y^{'}})^2=3^2 \\ x^2+\frac{x^2}{(y^{'})^2}=9\\ x^2(y^{'})^2+x^2=9(y^{'})^2\\ \\ (x^2-9)(y^{'})^2+x^2 = 0
Therefore, the required differential equation is (x^2-9)(y^{'})^2+x^2 = 0

Posted by

Gautam harsolia

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