...since m falls on both the ellipse and the line On, it will fall at n, aud Om = On - OP. THEOREM XIII. The sum of the distances from any point on the ellipse to the two foci is constant and equal to the major axis. Take P any point on the ellipse ; we are to prove...

...the line through the foci as the ж-axis, and the point midway between the foci as origin. Let 2 a = the sum of the distances from any point on the ellipse to the foci. Let F(c, 0) and F'(— c, 0) be the two foci. Let P(x, y) be any point on the locus. Then FP + F'P...

...points of which are so located with respect to two points within the curved line, called the foci, that the sum of the distances from any point on the ellipse to the foci is constant or equal for any one ellipse. Referring to the figure, the combined length of the two lines...

...(point of intersection of the major and minor axes), which points it is his custom to call 'foci'93; and the sum of the distances from any point on the ellipse to these foci is always equal to the major axis. It follows that the distance from the points of the ellipse...

...foci and the distance from the origin to b. You already know these distances as b and c, and you know that the sum of the distances from any point on the ellipse to the foci is 2fl. The point (0, b) on the y-axis is halfway between the foci, and the sum of its distances from...

...ellipse (figure 6.10). The position of each tack is called a focus. Since the string is not elastic, the sum of the distances from any point on the ellipse to the two foci is always the same. Kepler placed the Sun at a focus of the ellipse. His calculations showed...

...focus of the ellipse3. 3For the ellipse (j/a)2 + (y/b)2 = 1 the foci are the points (±\/a2 - 62,0). The sum of the distances from any point on the ellipse to the two foci is a constant. Different values of e giving different shapes of ellipse, with the special...

...from X to the center of the earth is equal to the semimajor axis, a = ^(r} + r2). (This is because the sum of the distances from any point on the ellipse to the two foci must equal 2a = rl + r2, and X is equidistant from the two foci, one of which is at the center...

...L. From symmetry, we have [cf. Stein (1977)] LL I rds= f r'ds. (3.100) oo By virtue of the property that the sum of the distances from any point on the ellipse to the two foci is constant and equal to the major axis, we get L r LL -| LL frds=- frds+fr'ds = - f (r +...

...J. We know that BBl , the minor axis, is a line of symmetry in the ellipse; so: BF=BF1. We also know that the sum of the distances from any point on the ellipse to the two foci is always the same. The A sum of the distances from B is BF+BF1; from A it is AF+AFl. So:...