# Hyperbola. Propositions and instructions. Propositions 6-8.

- Author:
- Alla Stolyarevska

- Topic:
- Hyperbola

**Proposition 6.**The axes of the canonical coordinate system are the symmetry axes of the hyperbola x² / a² - y² / b² = 1 , and the origin of the coordinate system is its center of symmetry.

**Conclusions:**MN=NM1, MA=AM2, and MO=OM’ for an arbitrary point M, therefore the ellipse has the symmetry axes and the center of symmetry.

*Supplementary problems:*

**Proposition 7.**Show that if a point moves along the hyperbola so that absolute value of its abscissa increases indefinitely, then the distance from point to one of the asymptotes has zero as its limit.

**Proposition 8.**Show that the distances r1, r2 from an arbitrary point M(x,y) on the hyperbola to each of the focuses dependent on its abscissa in the following manner: r1=F1M=a-e*x; r2=F2M=a+e*x , where e - eccentricity, a - the major semiaxis.