Can you find nice parameters for a point that has two closest points on a quadratic function?

Question

When you have a quadratic function

$$f(x) = a x^2 + bx + c$$

and a point

$$P = (x_p,y_p)$$

you can find one or two points on the graph of $f$ that have minimum (euclidean) distance

$$d_{P,f}(x) = \sqrt{(x-x_p)^2 + (f(x)-y_p)^2}$$

Example

When you have $f(x) = x^2$ (so $a=1, b = c = 0)$ and $x_p= 0, y_p > 5$ you can see that you will have two solutions. But I don't know how to compute them exactly (not with an approximation).

You could use $((d_{f,P} (x))^2)' \stackrel{!}{=} 0$, which is a polynomial of degree 3 and thus exactly solvalble. But I would like to get to know some $a, b, c, x_p, y_p$ for which there are two solutions and the parameters should be "nice" (integers, if possible or at least short expressions). Is this possible?

Solution

Here is an image of $f(x) = x^2-1$ with $P = (0,4)$ which has the solutions $x_{1,2} = \pm \sqrt{\frac{9}{2}}$:

Answer 1

Take $a = 1 , b = -4 $ and $c = 3$

Take the point to be $ (x_p,y_p) = (2,4) $

In this case you will get exactly two solutions .

Hint : The catch is that if you take a quadratic equation the line $x = x_p$ where $x_p$ is the $x$ co-ordinate of the minima of the quadratic will contain points which will give two solutions for minimum distance .

Just draw a rough graph from the parameters I have mentioned , you will get it .

Answer 2

as you note in your question, the general case will require solving a cubic equation. btw it may simplify the calculation if you translate the coordinate system so that your point $P$ is the origin. (it is not difficult to obtain the transformed values of $a, b$ and $c$).

when the cubic has three real roots, two will correspond to minimum distances, and the third will be a local maximum.

when you have one real and two conjugate complex roots, the real root will be a minimum distance.

the parabolic graph of $f(x)$ separates $\mathbb{R}^2$ into two open regions where the two cases apply - intuitively one region lies "inside" the curve, and the other consists of points "outside" the curve.