sign in sign up
<<
Home , Ellipse

An ellipse is an -shaped that has the appearance of an elongated . Ellipse

An ellipse is the set of points in a for which the sum of the distances from two fixed points is a given constant. The two fixed points are the focal points of the ellipse; the passing through the focal points is called the axis. The points of intersection of the axes and the ellipse are called the vertices. y •P • P(x,y) • F• F• • vertex vertex • F (-c,0) • •F(c,0 ) • vertex x focal focal axis focal focal point point point point There are two standard position ellipses:

 An ellipse with axis along the x-axis of the xy plane with origin midway between the focal points with coordinates F(c, )0 and F ' (−c, )0 and vertices V (a, )0 and V ' (−a )0,

 An ellipse with axis along the y-axis of the xy plane with origin midway between the focal points with coordinates F ,0( c) and F ' ,0( − c) and vertices V ,0( a) and V ' ,0( − a) The distance between the two focal points is 2c

The distance between the two vertices on the major axis is 2a where a > c > 0 The distance between the two vertices on the minor axis is 2b

y

(0,b) This ellipse is in standard position • with major axis along the x-axis. The major axis of an ellipse intersects b a the foci and the minor axis does not. (-a, 0) • • • • (a, 0) The minor axis is the short axis. (-c, 0) c (c, 0) x An ellipse in standard position with major axis along the x-axis has • : (0, -b) x2 y 2 + =1 a2 b2 y (0, a) • (0, c) •

c a • • x (-b,0) b (b, 0) This ellipse is in standard position with major axis along the y-axis. (0, -c) • The major axis of an ellipse intersects the foci and the minor axis does not. • The minor axis is the short axis. (0, -a) An ellipse in standard position with major axis along the y-axis has equation:

y 2 x2 + =1 a2 b2 Standard Position Ellipses

An ellipse with focal points (c, 0) and (-c, 0) and vertices (a, 0) and (-a, 0), where a > c > 0, is in standard position with axis along the x-axis and has equation

x 2 y 2 + = 1 where b = a 2 − c 2 a 2 b 2

Similarly, an ellipse with focal points (0, c) and (0, -c) and vertices (0, a) and (0, -a), where a > c > 0, is in standard position with axis along the y-axis and has equation

y 2 x 2 + = 1 where b = a 2 − c 2 a 2 b 2 Sketch the graph of the ellipse with equation 9x2 +16 y2 =144 and find its focal points.

We must get this ellipse in standard form, so we must divide both sides by 144 so that the right side of the equation is 1. x2 y 2 + =1 16 9 Since the coefficient in the denominator of x 2 is larger than the coefficient in the denominator of y 2 , the ellipse is in standard position with axis along the x-axis a2 =16 and b2 = 9 Axis intercepts: (4, 0), (-4, 0) and a = 4 and b = 3 (0, 3), (0, -3) Now, use the fact that b = a 2 − c 2 to find the value of c. b = a2 − c2 Focal Points: 2 2 2 − b = a − c ( )0,7 and ( )0,7 b2 − a2 = −c2 a2 − b2 = c2 a2 −b2 = c 42 − 32 = c 16 − 9 = c ± 7 = c Graphing: Since there are no constants being added or subtracted from x or y there are no vertical or horizontal shifts. So the ellipse is in standard position with center at the origin. Sketch the graph of the ellipse with equation 25 x2 + 4y2 =100 and find its focal points. Find an equation of the ellipse in standard position that has a vertex at (5,0) and a focal point at (3,0).

Since they say it is in standard position the center is at the origin. they tell you that the vertex is at (5,0), so this tells you that the major axis is aligned with the x-axis.

The other vertex is at (-5,0) and the other focal point is (-3,0)

This tells us that a = 5, and c = 3. Using the formula b = a2 − c2 we find that b = 4 The formula that must be used is when a 2 is under x2 x2 y 2 + =1 By substitution we get… a2 b2 x2 y2 + =1 or 16 x2 + 25 y2 = 400 25 16

Sketch together Find the equation of the ellipse in standard position that has vertex at (0, 6) and focal point at (0, 2). Sketch the graph. Sketch the graph of the ellipse and find its focal points and vertices.

9x2 − 72 x + 4y2 +16 y +124 = 0

Complete the square on both terms x and y. (9 x2 −8x) + (4 y2 + 4y) = −124 (9 x2 −8x +16 ) − 16(9 ) + (4 y 2 + 4y + )4 − )4(4 = −124 (9 x − )4 2 + (4 y + )2 2 = 36 (x − )4 2 (y + )2 2 + =1 4 9 The dome of the Mormon Tabernacle in is 250 feet long, 150 feet wide, and 80 feet high, and its longitudinal is in the of an ellipse. The conductor for performances stands at one end of the focal points of the ellipse, and recording equipment can be placed at the other. In this way sound heard by the conductor corresponds very closely to the sound being recorded. Determine the location of these points.

this is the situation when an xy-plane (0,80) is superimposed on a cross section of the dome. The width of the dome plays no part in the calculations.

since the ellipse is positioned this way (-125,0) (125,0) we will use the equation for this standard position: x2 y2 + =1 a2 b2 We will use the formula b = a 2 − c 2 and substitute in our values for a and b to find c.

80 = 125 2 − c2 c ≈ 96 feet

So, the conductor should stand 125 – 96 = 29 feet from one end of the building, and the recording equipment should be placed an equal distance from the other. The eccentricity of an ellipse tells how the ellipse differs from a circle.

c a2 − b2 eccentrici ty = = for standard position ellipses a a

Note: as the focal point approaches the origin, the ellipse approaches a circle, and the eccentricity approaches 0. The ellipse becomes increasingly elongated as the focal points approach the vertices which occurs when the eccentricity approaches 1.