Tiêu đề 62 Conic Sections - Ellipses.pdf ✅

MSTC 62: Conic Sections – Ellipses 1. Locus An ellipse is the locus of points (x, y) with a fixed sum of distances from two points called the foci (singular, focus). 2. Equations From the center-radius form of a circle, (x − h) 2 + (y − k) 2 = r 2 (x − h) 2 r 2 + (y − k) 2 r 2 = 1 In circles, the denominators of the square terms are the same. However, when the denominators are different, they represent the widths of the oval in that direction. Consider a > b, and (x − h) 2 a 2 + (y − k) 2 b 2 = 1 The graph this equation represents is an ellipse that is wider in the x - direction. In other words, it is a horizontal ellipse. Now, if a is under the y square term, (x − h) 2 b 2 + (y − k) 2 a 2 = 1 The graph this equation represents is an ellipse that is wider in the y - direction. In other words, it is a vertical ellipse. The earlier equations are called the center form of the equation of ellipse. Expanding the terms, say for the horizontal ellipse (the final form also applies to vertical ellipses), b 2 (x − h) 2 + a 2 (y − k) 2 = a 2b 2 b 2 (x 2 − 2hx + h 2 ) + a 2 (y 2 − 2ky + k 2 ) = a 2b 2 b 2x 2 − 2b 2hx + b 2h 2 + a 2y 2 − 2a 2ky + a 2k 2 = a 2b 2 b 2x 2 + a 2y 2 − 2b 2hx − 2a 2ky + b 2h 2 + a 2k 2 − a 2b 2 = 0 Let A = b 2 , C = a 2 ,D = −2b 2h, E = −2a 2k, and F = b 2h 2 + a 2k 2 − a 2b 2 , Ax 2 + Cy 2 + Dx + Ey + F = 0 This form is called the standard equation of the ellipse. An equation that resembles the standard form can be verified as an ellipse if the coefficients of the terms have the same sign. 3. Properties For ellipses expressed in their center form, Characteristic Definition Value Center The point of intersection of the major and minor axes. It is also the centroid of an ellipse. (h, k) Semi-major Axis The distance from the center to a vertex. The length of the major axis is 2a. a Semi-minor Axis The distance from the center to a co- vertex. The length of the major axis is 2b. b
Vertices The corner points at which the ellipse makes its maximum turn. (h ± a, k) if horizontal (h, k ± a) if vertical Co-vertices The corner points at which the ellipse makes its minimum turn. (h, k ± b) if horizontal (h ± b, k) if vertical Focal Distance The distance from the center to a focus. The distance between the foci is 2c. c = √a 2 − b 2 Eccentricity The ratio of the distance of the focal distance to the vertex distance. The eccentricity of an ellipse is always less than 1. e = c a Directrix A line parallel to the latus rectum of the ellipse, perpendicular to its major axis. d = a e = a 2 c Latus Rectum (plural, latera recta) A segment that passes through the foci and is perpendicular to the major axis, which intersects the conic. LR = 2b 2 a Note that for a circle, since a = b = r, then c = 0 and e = 0. 4. Area and Perimeter The area of an ellipse is given exactly by A = πab Where a and b are the lengths of the semi-major and semi-minor axes, respectively. Graph of a horizontal ellipse Graph of a vertical ellipse