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Homework Statement
Using the method of lagrange multipliers, find the points on the curve 3x² - 4xy + 6y² = 140 which are closest and furthermost from the ORIGIN and the corresponding distances between them
The Attempt at a Solution
I have done roughly half the question but appear to be stuck!
Firstly I find distance from a point (x,y) to the (0,0) using distance equation.
This yields:
d² = x² + y² = f(x,y)
My constraint equation is given by the equation of the curve given, i.e.
g(x,y) = 3x² - 4xy + 6y² = 140
Now I want to use grad(f) = λgrad(g)
Therefore;
2xi + 2yj = λ[(6x-4y)i + (-4x+12y)j]
And by equating coefficients,
2x = λ(6x-4y) (1)
2y = λ(-4x+12y) (2)
This is where I get stuck, I have never encountered a problem like this whereby one of the variables depends on itself. What is the best way to tackle this? Should I sub (1) into (2)?