Minimum distance between $x = -y^2$ and $(0,-3)$

Question

Find the minimum distance from the parabola $x + y^2 = 0$ (i.e. $x = -y^2$) to the point $(0,-3)$.

This is a homework question. When I try to use the derivative and substitute $-y^2$ for $x$, I get a nonsensical expression (if evaluated, I end up with an imaginary value for $y$ when trying to calculate the distance). What is the proper way to solve this question?

Answer 1

Let $(-y_1^2,y_1)$ be the point on the parabola, which is at the minimal distance from $(0,-3)$, i.e., $$R(y_1) = r(y_1)^2 = y_1^4 + (y_1+3)^2$$ should be minimum. Can you now find out $y_1$ by setting the derivative of $R(y_1)$ to $0$?

Move the mouse over the gray area for the answer.

$$\dfrac{dR}{dy_1} = 4y_1^3 + 2y_1 + 6$$ which gives us clearly $y_1=-1$ as one solution and the remaining roots are complex. Hence, the point on the parabola is $(-1,-1)$ and is at a distance $\sqrt5$.