Find the slope of the curve x^2 + xy + y^2 = 7 at (1,2)

To do this problem we first apply the product
rule:

d ( xy) / dx = y + x*dy/dx

to find the derivative of x^2 + xy + y^2 =
7.

So the derivative of x^2 + xy + y^2 = 7
is

2x + x*dy/dx +y + 2y* dy/dx
=0

Now taking dy/dx to one
side,

=> dy/dx( x +2y ) = -x^2 -xy -
y^2

=> dy/dx = (-x^2 -xy - y^2) /
(x+2y)

At (1,2)

dy/dx = (-x^2
-xy - y^2) / (x+2y)

=>( - 1^2 - 1*2 - 2^2)/ (1+
4)

=> (-2 -2 -4
)/5

=>
-8/5

Therefore the required slope is
-8/5