Intersection points of an ellipse with the coordinates axes. To find the intersection points of an ellipse with the x-axis, we must solve the system of equationsWe obtain two vertices of the ellipse: М 1(–а; 0), М 2(а; 0), is called the major axis of the ellipse; а is the major semi axis We find the intersection of the ellipse with the y -axis by solving the system We obtain the two other vertices of the ellipse, М 3(0;– b) and М 4(0; b). is called the minor axis of the ellipse, and b is the minor semi axis. It is seen from equation (17) and the figure that the ellipse is symmetric with respect to the axes Ox and . The eccentricity and directrix of an ellipse. Consider the focal radii of an ellipse ; . By definition, we have . Consider the difference of squares ; , or . y d1 М(х;у) d2 r1 r2 F1(–c; 0 ) 0 F2(c; 0 ) x
x =– l x = l
to determine the focal radii, we solve the system of equations or Definition. The ratio of distances between the foci to the sum of focal radii is called eccentricity: . If the distance between the foci is less than 2 а, then the eccentricity is Thus, the focal radii of the ellipse are , . Definition. The directrix of an ellipse is the straight line parallel to the y -axis such that the ratio of the focal radius to the distance from an ellipse point to it is constant and equal the eccentricity. Let us draw two straight lines x=–l and x=l parallel to the y -axis and find l such that the ratio of the focal radius to the distance from a point М to this straight line is constant and equals the eccentricity: . Substituting the distance and the focal radius, we obtain . The ratio is equal to the eccentricity when , i.e., is the directrix. By analogy, we obtain equations of the directrices: ; , where .
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