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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's suppose the centre is at $(0,b)$
Now, the equation of a circle with centre $(0,b)$ and radius$ = 3$ units
$(x-0)^2+(y-b)^2=3^2----(i)$

$x^2+y^2+b^2-2 y b=9$

Now, differentiate w.r.t $x$, we get,

$2 x+2 y y^{\prime}-2 b y^{\prime}=0$

$2 x+2 y(y-b)=0$

$(y-b)=\frac{-x}{y^{\prime}}$

Put the value from equation (ii) in (i)

$(x-0)^2+\left(\frac{-x}{y^{\prime}}\right)^2=3^2$

$x^2+\frac{x^2}{\left(y^{\prime}\right)^2}=9$

$x^2\left(y^{\prime}\right)^2+x^2=9\left(y^{\prime}\right)^2$

$\left(x^2-9\right)\left(y^{\prime}\right)^2+x^2=0$

Therefore, the required differential equation is $\left(x^2-9\right)\left(y^{\prime}\right)^2+x^2=0$

Posted by

Gautam harsolia

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