THE ELLIPSE Week 17

THE DEFINITION The locus of a point that moves in such a way that the ratio of its distances from a fixed point S (the focus) and from a fixed line ZL (the directrix) is constant (e) and less than 1

BASIC FEATURES The fixed line ZL is perpendicular to the x-axis Let A divide SZ internally in the ratio e:1 (e<1) Let A’ divide SZ externally in the ratio e:1 (e<1)

BASIC FEATURES Thus, A and A’ are members of the loci of the ellipse (similar to the point P) O is the origin and the mid-point of AA’

THE FOCUS Let AA’ = 2a SA = e. AZ SA’ = e. A’Z SA’ - SA = e (A’Z – AZ) (OS+a) – (a-OS) = e(2a) 2.OS = 2.ae OS = ae The focus, S is the point with coordinates (-ae, 0)

EQUATION OF THE DIRECTRIX SA = e. AZ SA’ = e. A’Z SA’ + SA = e (A’Z + AZ) (2a) = e[(a+OZ) + (OZ-a)] 2a = 2e. OZ OZ = a/e

THE EQUATION OF AN ELLIPSE

PROPERTIES OF AN ELLIPSE Symmetrical about both axes The foci are S (-ae, 0) and S’ (ae, 0) The lines x = -a/e and x = a/e are the directrices

PROPERTIES OF AN ELLIPSE AA’ = 2a (major axis) BB’= 2b (minor axis) O is the center e is the eccentricity

ELLIPSE: ALTERNATIVE DEFINITION An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points is constant. The two fixed points are called the foci

EQUATION OF THE ELLIPSE The equation of an ellipse can also be found by using the distance formula and the definition An ellipse has foci at (5, 0) and (-5, 0). The distances from either of the x-intercepts to the foci are 2 units and 12 units. Find the equation of the ellipse.

EQUATION OF THE ELLIPSE Consider the x-intercept, A. AS+AS’=14 units The sum of the distances from any point P(x, y) to the two foci must also be 14 units. The distance formula can be used to find the equation of the ellipse -5,0 5,0

EQUATION OF THE ELLIPSE

Properties of Ellipses An ellipse is the set of all points in a plane such that the sum of the distances from the foci is constant An ellipse has two axes of symmetry The axis of the longer side of the ellipse is called the major axis and the axis of the shorter side is the minor axis The focus points always lie on the major axis The intersection of the two axes is the center of the ellipse Major Axis Focus Center Minor Axis Focus

PROPERTIES OF THE ELLIPSE The sum of the distances from any point to the foci = 2a (the length of the major axis). The distance from the centre to either foci = c units

Important Information - Ellipses Equation of the Ellipse Foci Points Is the major axis horizontal or vertical? Center of the Ellipse (x – h)2 + (y – k)2 a2 b2 ( h + c, k) and (h – c, k) Horizontal (h, k) b2 a2 (h, k + c) and (h, k – c) Vertical = 1 = 1 Important Notes: In the above chart, a2 > b2 always so a2 is always the larger number If the a2 is under the x term, the ellipse is horizontal, if the a2 is under the y term the ellipse is vertical You can tell that you are looking at an ellipse because: x2 is added to y2 and the x2 and y2 are divided by different numbers (if numbers were the same, it’s a circle)

Worked Example Given an equation of an ellipse 16y2 + 9x2 – 96y – 90x = -225 find the coordinates of the center and foci as well as the lengths of the major and minor axis. Then draw the graph. 16 (y2 – 6y + o) + 9 (x2 – 10x + o) = -225 + 16 (o) + 9(o) 16 (y2 – 6y + 9) + 9 (x2 – 10x + 25) = -225 + 16(9) + 9(25) 16 (y – 3)2 + 9 (x – 5)2 = 144 (y – 3)2 + (x – 5)2 9 16 = 1 Center: (5, 3) 16 > 9 so the foci are on the horizontal axis c = 16 – 9 c = 7 Foci: ( 5 + 7, 3) and (5 – 7, 3) Major Axis Length = 4 (2) = 8 Minor Axis Length =

Sample Problems For 49x2 + 16y2 = 784 find the center, the foci, and the lengths of the major and minor axes. Then draw the graph. Write an equation for an ellipse with foci (4, 0) and (-4, 0). The endpoints of the minor axis are (0, 2) and (0, -2). 20 4 X2 + y2 Foci: (0, - 33) (0, 33) Center: (0, 0) Length of major= 14 Length of minor= 8 = 1 #1