Question

Question #d6ef5

Answer 1

The differential equation for the family of circles is:

`dy/dx = (r+x)/(r-y)` where `r>0`.

Explanation:

The general equation of a circle with centre `(a,b)` and radius `r` is:

`(x-a)^2 + (y-b)^2 = r^2`

If we want the circle in the second quadrant then we require the centre `(a,b)` to lie on the line `y=-x` so that `a=-b` and `r=b` with `r>0`, which gives us:

`(x+r)^2 + (y-r)^2 = r^2 \ \ \ \ \ \ \` where `r>0`

Differentiating wrt `x` we get:

`2(x+r) + 2(y-r)dy/dx = 0`
`:. (x+r) + (y-r)dy/dx = 0`
`:. (y-r)dy/dx = -(x+r)`
`:. dy/dx = (r+x)/(r-y)`

So the differential equation for the family of circles is:

`dy/dx = (r+x)/(r-y)` where `r>0`.