Finding the minimal distance between a parabola and a point without using calculus

Question

The smallest distance between the origin and a point on the parabola $y=x^2-5$ can be expressed as $\dfrac{\sqrt{a}}b$, where $a$ is not divisible by the square of any prime. Find $a+b$.

I have been browsing the web, and I can't find any methods that don't use Calculus. Help is greatly appreciated.

Answer 1

The given parabola can be parameterized as follows

$$\left\{ (t,t^2-5) \mid t \in \mathbb R \right\}$$

The squared Euclidean distance from a point on the parabola to the origin is

$$t^2 + (t^2 - 5)^2 = \cdots = t^4 - 9 t^2 + 25 = \cdots = \left( t^2 - \frac 92 \right)^2 + \frac{19}{4}$$

The minimal squared Euclidean distance is $\frac{19}{4}$, which is attained at $t = \pm \frac{3}{\sqrt{2}}$.

Answer 2

Hint...the distance $d$ from the origin to any point on the parabola is given by $$d^2=x^2+(x^2-5)^2$$

So rather than use calculus you can complete the square to find the minimum $d$