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  • Form the differential equation of the family of circles touching the y-axis at origin.

6.    Form the differential equation of the family of circles touching the y-axis at origin.

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If the circle touches y-axis at the origin then the centre of the circle  lies at the x-axis 
Let r be the radius of the circle
Then, the equation of a circle with centre at (r,0) is
(x-r)^2+(y-0)^2 = r^2
x^2+r^2-2xr+y^2=r^2\\ x^2+y^2-2xr=0    -(i)
Now, differentiate w.r.t x
2x+2y\frac{dy}{dx}-2r=0\\ y\frac{dy}{dx}\Rightarrow yy^{'}+x-r=0
yy^{'}+x=r    -(ii)
Put equation (ii) in equation (i)
 x^2+y^2=2x(yy^{'}+x)\\ y^2=2xyy^{'}+x^2
Therefore, the required equation is   y^2=2xyy^{'}+x^2

Posted by

Gautam harsolia

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