Question

Choose the correct option from the given alternatives:

x² + y² = a² is a solution of

Options

• `("d"^2"y")/"dx"^2 + "dy"/"dx" - "y" = 0`

• y = x`sqrt(1 + ("dy"/"dx")^2) + "a"^2 "y"`

• y = x`"dy"/"dx" + "a" sqrt(1 + ("dy"/"dx")^2)`

• `("d"^2"y")/"dx"^2 = ("x + 1")"dy"/"dx"`

Answer 1

y = x`"dy"/"dx" + "a" sqrt(1 + ("dy"/"dx")^2)`

Hint:

x² + y² = a²     ∴ 2x + 2y`"dy"/"dx" = 0`

∴ `"dy"/"dx" = - "x"/"y"`

∴ `"x" "dy"/"dx" + "a" sqrt(1 + ("dy"/"dx")^2)`

`= "x"(- "x"/"y") + "a"sqrt(1 + "x"^2/"y"^2) = - "x"^2/"y" + "a" xx "a"/"y"`

`= ("a"^2 - "x"^2)/"y" = "y"^2/"y" = "y"`