Minimize a function in two variables with constraint

Question

I have to minimize this: \begin{align*} \min&\quad{ (x-3)^2+(y-1)^2} \\ s.t.& \quad 2x+y \leq 2 \\ &\quad x^2 + 2y = 3\\ &\quad x, y \geq 0 \end{align*} Can I isolate $y$ in the second constraint and substitute it in the first?

Answer 1

Let $$(x-3)^2+(y-1)^2=z~~~(1)$$ Let us put $x^2=(3-2y)$ in (1), we get $$y^2-4y-6x+13-z=0.~~~(2)$$ $z$ will attain optimum value if when the line $y=2-2x$ touches the curve (2) which is a parabola. Let us but this line in (2) $$4x^2-6x+9-z=0.~~~~(3)$$ Now demand $B^2=4AC$.This gives $z=27/4,$ the answer.