Find the differential equation of all circles having radius 9 and centre at point (h, k).

#### Solution

Equation of the circle having radius 9 and centre at point (h, k) is

(x - h)^{2} + (y - k)^{2} = 81, .....(1)

where h and k are arbitrary constant.

Differentiating (1) w.r.t. x, we get

`2("x - h") * "d"/"dx" ("x - h") + 2 ("y - k") * "d"/"dx" ("y - k") = 0`

∴ (x - h)(1 - 0) + (y - k)`("dy"/"dx" - 0) = 0`

∴ (x - h) + (y - k) `"dy"/"dx" = 0` .....(2)

Differentiating again w.r.t. x, we get

`"d"/"dx" ("x - h") + ("y - k") * "d"/"dx"("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("y - k") = 0`

∴ `(1 - 0) + ("y - k") ("d"^2"y")/"dx"^2 + "dy"/"dx" * ("dy"/"dx" - 0) = 0`

∴ `("y - k") ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2` + 1 = 0

∴ `("y - k") ("d"^2"y")/"dx"^2 = - [("dy"/"dx")^2 + 1]`

∴ `"y - k" = (- ("dy"/"dx")^2 + 1)/(("d"^2"y")/"dx"^2` ....(3)

From (2), x - h = - (y - k)`"dy"/"dx"`

Substituting the value of (x - h) in (1), we get

`("y - k")^2 ("dy"/"dx")^2 + ("y - k")^2 = 81`

∴ `("dy"/"dx")^2 + 1 = 81/("y - k")^2`

∴ `("dy"/"dx")^2 + 1 = (81 * ("d"^2"y")/"dx"^2)/[("dy"/"dx")^2 + 1]^2`

∴ `81 (("d"^2"y")/"dx"^2)^2 = [("dy"/"dx")^2 + 1]^3`

This is the required D.E.