Hello? I want to know a sum of distances from two points to a point on a circle.
There is a unit circle $C$ on a $3$D plane. For two given points $A = (2,2,2), B =(-2,3,4)$.
Let $P$ be a point in $C$.
What is a mimimum or maximum value of $\overline{PA} + \overline{PB}$ ?
I tried some methods. It is not useful to consider some ellipes and differentiation.
Rotating points, reflection, inequality.. I cannot find a solution.
Please give me a solution. Thank you.
you do not specify how the unit circle $C$ is positioned in $\mathbb R^3$: for example, is it $x^2 + y^2 = 1$, and $z = 0$, or $y^2 + z^2 = 1$ and $x = 0$? In fact, it need not even be coplanar with any of the coordinate planes. All I can infer is that it has unit radius, and is (probably) centered at the origin.
That said, suppose it is parametrized by $$C : [0,2\pi) \to \mathbb R^3, \quad P(x,y,z) = (\cos \theta, \sin \theta, 0).$$ Then the sum of distances as a function of $\theta$ is given by $$PA + PB = \sqrt{13-4(\cos \theta + \sin \theta)} + \sqrt{30 + 4 \cos \theta - 6 \sin\theta}.$$ However, the computation of the derivative and solution for critical points is not tractable in elementary closed form. The best you can do is numerical approximation. The maximum distance corresponds to approximately $$\theta \approx 4.5694009600354164241$$ radians, and the minimum distance corresponds to $$\theta \approx 1.3446037205849027071$$ radians. The maximum distance is approximately $10.133951033365217067$ and the minimum is approximately $7.8693935495850818355$.
