Question

Find the equation of the tangent line to the curve 2exy=x+y at (0,2).

Answer 1

Answer:

The equation of the tangent line to the curve

3 x - y = 2

Step-by-step explanation:

Step(i):-

Given function  = f(x,y) = $2e^{xy} -x-y=0$   ...(i)

Differentiating equation (i) with respective to 'x' , we get

$2 e^{xy} \frac{d}{d x} (x y) -1 -\frac{dy}{dx} =0$

apply  formula

$\frac{d}{dx} (UV) = UV^{l} +V U^{l}$

step(ii):-

⇒   $2 e^{xy} (x(\frac{d}{d x} ( y))+y(1)) -1 -\frac{dy}{dx} =0$

⇒ $2 e^{xy} (x(\frac{d}{d x} ( y))+ 2e^{xy} y(1)) -1 -\frac{dy}{dx} =0$

Taking common d y/d x

$(2 e^{xy} (x) -1)\frac{dy}{dx} =1- 2e^{x y} y(1))$

$\frac{dy}{dx} =\frac{1- 2e^{x y} y(1))}{(2 e^{xy} (x) -1)}$

put At (0,2)

$\frac{dy}{dx} =\frac{1- 2e^{0} 2(1))}{(2 e^{0} (0) -1)}=\frac{1-4}{-1} =3$

slope of the curve m = 3

Step(iii):-

The equation of the tangent line to the curve

$y-y_{1} =m(x-x_{1} )$

y - 2 = 3 ( x - 0 )

3 x - y = 2

Final answer:-

The equation of the tangent line to the curve

3 x - y = 2