f(x,y) = d² = x² + y²

g(x,y) = 3x²-4xy+6y² = 140

Then by my calculations,

grad(f) = 2xi+ 2yj

grad(g) = (6x-4y)i+ (-4x+12y)j

Then grad(f) = λgrad(g) produces these equations

2x = λ(6x-4y) (eq.1)

2y = λ(-4x+12y) (eq.2)

rearranging 1 gives:

λ = 2x/(6x-4y) (eq.3)

rearranging 2 gives:

λ = 2y/(-2x+6y) (eq.4)

then by cross multiplying,

-2x² + 6xy = 3xy - 2y²

taking all terms to left hand side,

-2x² +3xy + 2y² = 0

(-2x - y)(x - 2y)=0

therefore x = y/-2 or x = 2y

This is where I am stuck, now I have a relationship between x and y, what do I do next??