\f\f\fGiven function is f x, y x x2 y 2 and we have to draw level curves correponding to different c values. Here note that the given function is not defined for 0,0 as f 0,0 0 0 is not defined. 0 0 0 The level curves for the given function is defined by setting f x, y c where c is any constant 2 2 Here the level curves are not defined for c 1,1. Because if we have c 1,1 then x f x, y c c x2 y 2 x2 c 2 x 2 c 2 x 2 c 2 y 2 c 2 1 x 2 c 2 y 2 0 2 2 x y cy c 2 1 x, as c 1,1 c 2 1 c 2 1 0 c 2 1 is not a real number So we cannot draw level curve for c 1,1. Here we can draw level curves for c 1,1. Set c 1, 0.25,0,0.25,1 in the given equation c f x, y to get level curves as shown below. Hence the equation of level curves are f x, y 1, f x, y 0.5, f x, y 0, f x, y 0.5, f x, y 1 x Now f x, y 1 x2 y 2 x Here note that x y 2 2 1 x2 1 x 2 x 2 y 2 y 2 0 y 0, the x-axis x2 y 2 1 x x 2 y 2 0 so the level curve f x, y 1 is defined by negative x-axis where x 0 that is y 0, x 0. x f x, y 0.5 Since x x y 2 2 x2 y 2 f x, y 0.5 x x y 2 f x, y 1 Here note that 2 x2 0.25 x 2 0.25 x 2 0.25 y 2 0.25 y 2 0.75 x 2 y 3 x x2 y 2 0.5 x 0.5 x 2 y 2 x 0, so the corresponding level cuve is y 3 x, x 0 x f x, y 0 Since x2 y 2 0.5 0 x0 x x2 y 2 0.5 x2 0.25 x 2 0.25 x 2 0.25 y 2 0.25 y 2 0.75 x 2 y 3 x x2 y 2 0.5 x 0.5 x 2 y 2 x 0, so the corresponding level cuve is y 3 x, x 0 x2 1 2 1 x 2 x 2 y 2 y 2 0 y 0, the x-axis 2 2 2 x y x y x x x y 2 2 1 x x 2 y 2 0 so the level curve f x, y 1 is defined by positive x-axis where x 0 that is y 0, x 0. Given function is f x, y x x2 y 2 and we have to draw level curves correponding to different c values. Here note that the given function is not defined for 0,0 as f 0,0 0 0 is not defined. 0 0 0 The level curves for the given function is defined by setting f x, y c where c is any constant 2 2 Here the level curves are not defined for c 1,1. Because if we have c 1,1 then x f x, y c c x2 y 2 x2 c 2 x 2 c 2 x 2 c 2 y 2 c 2 1 x 2 c 2 y 2 0 2 2 x y cy c 2 1 x, as c 1,1 c 2 1 c 2 1 0 c 2 1 is not a real number So we cannot draw level curve for c 1,1. Here we can draw level curves for c 1,1. Set c 1, 0.25,0,0.25,1 in the given equation c f x, y to get level curves as shown below. Hence the equation of level curves are f x, y 1, f x, y 0.5, f x, y 0, f x, y 0.5, f x, y 1 x Now f x, y 1 x2 y 2 x Here note that x y 2 2 1 x2 1 x 2 x 2 y 2 y 2 0 y 0, the x-axis x2 y 2 1 x x 2 y 2 0 so the level curve f x, y 1 is defined by negative x-axis where x 0 that is y 0, x 0. x f x, y 0.5 Since x x y 2 2 x2 y 2 f x, y 0.5 x x y 2 f x, y 1 Here note that 2 x2 0.25 x 2 0.25 x 2 0.25 y 2 0.25 y 2 0.75 x 2 y 3 x x2 y 2 0.5 x 0.5 x 2 y 2 x 0, so the corresponding level cuve is y 3 x, x 0 x f x, y 0 Since x2 y 2 0.5 0 x0 x x2 y 2 0.5 x2 0.25 x 2 0.25 x 2 0.25 y 2 0.25 y 2 0.75 x 2 y 3 x x2 y 2 0.5 x 0.5 x 2 y 2 x 0, so the corresponding level cuve is y 3 x, x 0 x2 1 2 1 x 2 x 2 y 2 y 2 0 y 0, the x-axis 2 2 2 x y x y x x x y 2 2 1 x x 2 y 2 0 so the level curve f x, y 1 is defined by positive x-axis where x 0 that is y 0, x 0