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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 $(4,8)$.

algebra2/optimiz/distance/problemse.txt · Последние изменения: 2020/03/11 14:00 (внешнее изменение)