Find the differential equation of all circles touching the (i) x-axis at the origin (ii) y-axis at the origin

Explanation

(i)Let the equation of the circle be

`x^{2}+(y-a)^{2}=a^{2} `

`Rightarrow x^{2}+y^{2}-2 a y=0 `

`Rightarrow 2 x+2 y frac{d y}{d x}-2 a frac{d y}{d x}=0 `

`Rightarrow x+y-frac{d y}{d x}=a frac{d y}{d x} `

`Rightarrow a=frac{x+y frac{d y}{d x}}{frac{d y}{d x}}=frac{x+y y_{1}}{y_{1}}`


Hence, the required differential equation is

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