Hyperbola. Propositions and instructions. Propositions 6-8.

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.