Основное
Satellite page for ☞ Distance evaluation between geometric objects
1. Find the distance from the point $ (1,2,\dots,n) $ to the manifold $ x_1+x_2+\dots+x_n=1 $.
2. Find the set of the points in $ \mathbb R^{3} $ equidistant from the planes $$ x+2\,y+3\, z=0 \quad and \quad x+y+z=0 \ . $$
3. Find the coordinates of the point in the plane equidistant from the three given circles.
4. Find the distance from the point $ (5,6,7) $ to the nearest and the farthest points in the ellipsoid $$ \frac{x^2}{25}+\frac{y^2}{9}+\frac{z^2}{16}=1 \ . $$ Find the coordinates of these points.
5. Find the distance between the ellipses $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \quad and \quad \frac{x^2}{(a-h)^2}+\frac{y^2}{(b-h)^2}=1 \ , $$ and coordinates of the pair of their nearest points. Here $ 0\le h\le \min \{a_{},b\} $ .
6. [Bertrand]. Find the point providing the minimum value for the sum of the distances from it to two given lines and to a given point.
7. Ellipse $ (x-x_0)^2+1/4\,(y-y_0)^2=1 $ moves without rotation in the plane with its center tracing the curve $ x_0=t^2+t,y_0=t^2 $ for $ t\in [-1,3] $. Find the value for $ t_{} $ corresponding to the nearest position of the ellipse to the point $ (4,8) $.