Consider the following question:
The equation $73x^{2}$ + $72xy$ + $52y^{2} = 100$ defines an ellipse which is
centered at the origin, but has been rotated about it. Find the
semiaxes of this ellipse by maximizing and minimizing $f(x, y) = x^{2} +
y^{2}$ on it.
This problem seems very complicated to me. I'm self-studying multivariable calculus and I cannot figure out this problem. Let $g(x, y, z) = 73x^2 + 72xy + 52y^2 – 100$ and $f(x, y, z) = x^2 + y^2$
First, I computed $$g_{x} = 146x + 72y$$
$$g_{y} = 104y + 72x$$
$$g_{z} = 0 $$
$$f_{x} = 2x$$
$$f_{y} = 2y.$$
$$f_{z} = 0.$$
Then, I have
$$2x = \lambda (146x + 72y)$$
$$2y = \lambda(104y + 72x)$$
$$73x^{2} + 72xy + 52y^{2} = 100$$
I don't know how to proceed since this is a system of three variables. Also, this is my first lagrange multiplier attempt so I'm not confident about what to do next either. I've been using Paul's Online Notes to try and understand. My ultimate goal is to be able to apply this on inequalities because I've heard lagrange multipliers are helpful in math olympiad. This seemed like a good example.
Best Answer
This is a homogeneous problem and can be handled easily. Making $y = \mu x$ we have.
$$ \begin{cases} x^2+y^2 = x^2(1+\mu^2)\\ 73x^2+72 x y + 52 y^2=x^2(73+72\mu+52\mu^2) = 100 \end{cases} $$
so the problem is equivalent to
$$ \min_{\mu}f(\mu) = \frac{100(1+\mu^2)}{73+72\mu+52\mu^2} $$
now
$$ f'(\mu)=0\equiv (1+\mu)(16\mu-37) = 0 $$
and thus we have $\mu = -1$ and $\mu = \frac{16}{37}$ as stationary points etc.