Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Explanation

The equation of circle in the second quadrant which touch the coordinates axes are

`\left ( x+a \right)^{2}+\left ( y-a \right)^{2}= a^{2}\, a\varepsilon R---(i)`

where a is perimetric this equation contains ine arbitary constant so we shall differentiate it quce only and we shall get differentiate equation of first order.

Differentiating (i) w.r.t x, we get

`2\left ( x+a \right )+2\left ( y-a \right )\frac{dy}{dx}= 0`

`\Rightarrow a= -\left [ \frac{x+y\frac{dy}{dx}}{1-\frac{dy}{dx}} \right ]\Rightarrow a= \frac{x+p{y}}{p-1}\, where\, p= \frac{dy}{dx}`

substituting the value of a in (i), we get

`\left ( x+ \frac{x+py}{p-1} \right )^2+\left ( y-\frac{x+p{y}}{p-1} \right )^{2}= \left ( \frac{x+p{y}}{p-1} \right )^{2}`

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