Minimum Distance Problem with Calculus

Question

How do I find the minimum distance from the origin of a point $P$ in the first quadrant on the curve $0 = x^2 + y^2 − 2xy −8xy + 2015$.

This is what I have done so far or attempted. I assumed that the distance from the origin to the curve is given by

$$ d(x,y) = \sqrt{x^2+y^2} $$

and I want to minimize it subject to

$$ g(x)=x^2 + y^2 − 2xy −8xy + 2015 .$$

So I thought I could use the Lagrange multiplier method. But I don't know how to. Can someone please help me?

Answer 1

It is also possible to minimize the function $d(x,y)=x^2+y^2$, because if $\sqrt{x^2+y^2}$ is minimal then $x^2+y^2$ is also minimal. Now it is easy to use the method of Lagrange Multipliers.

Answer 2

for this case,another way is to use $2xy\le x^2+y^2$

$10xy=x^2+y^2+2015\le 5(x^2+y^2) \iff x^2+y^2 \ge \dfrac{2015}{4}$

if you want max ,$2xy\ge-(x^2+y^2)$