Plane Analytic Geometry, with Introductory Chapters on the Differential Calculus

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1915 edition. Excerpt: ...there. The two points A' and A where the transverse axis meets the ellipse are called the vertices of the ellipse. The curve can readily be described by fastening the ends of a string of length 2 a to pins inserted at the foci and pressing the point of the pencil against this string so as to keep it taut. The pencil can then slip along the string and describe a curve, which will be the desired ellipse, since the sum of the distances from the pencil-point to the two foci is always equal to the whole length of the string, 2 a. The curve is thus seen to have the shape indicated in Figure 46. It is clear that the shape of the curve (as distinguished from its size) depends only on the ratio of the length of the string to the distance between the foci. For instance, we shall have an ellipse of the same shape if we take a string three inches long and fasten its ends at points two inches apart as if we take a string three feet long and fasten its ends at points two feet apart. This ratio (4) e = =VZEE, a a Or, better still, by tying together the two ends of a string of length 2 (a + c) and placing the loop thus formed around the pins at the foci. which determines the shape of the ellipse is called its eccentricity. In the case of the circle it is zero; otherwise, it is a positive constant less than one. EXERCISES 1. Find the values of a. b, c, e for the ellipse--+ 1. F 25 16 What are the coordinates of its foci and vertices? 2. Find the equation of the ellipse whose foci are the points (±2, 0), and one of whose vertices is the point (3, 0). 3. The length of the major axis of an ellipse is 6. Find the equation of the ellipse and the coordinates of its foci if (a) e =, (6)'e = v5, 0)' e = 0. Draw the figure of the ellipse in each case. 4. What are the...