Ellipses in Co-ordinate Geometry
2 D Co-ordinate Geometry
Lecture-13
The ellipse
Dated:-09.05.2020
PPT-05
UG (B.Sc., Part-1)
Dr. Md. Ataur Rahman
Guest Faculty
Department of Mathematics
M.L. Arya, College, Kasba
PURNEA UNIVERSITY, PURNIA
The ellipse
Definition:
-
An ellipse is the
Locus of a point which
Moves in a plane such
That the ratio of its distance
from the fixed point (called focus)
and from the fixed line (called directrix)
is always constant and less than unity.
i.e.
Where S is called focus and AZ is called directrix of the
ellipse.
O
X
S(focus)
N
B
B´
M
Z
Directrix
F(focus)
Standard equation of an Ellipse
Let S (ae,0) be the focus and
be the eq. of the given directrix
ZM of the ellipse,where e
is the eccentricity.
is the eccentricity.
Let P(x,y) be any point on the
Ellipse, then by the definition
of the ellipse,
From
O
A(a,0
)
X
S(ae,0)
A´(-a,0)
P(x,y)
N
B
B´
M
Z
Proof Continue
From (1),We get
The ellipse is defined as the locus of a point moving in a plane such that the ratio of its distance from the focus to the directrix is constant and less than unity. This concept is explained through the standard equation of an ellipse and its proof, providing a deeper insight into this geometric shape.
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Presentation Transcript
2 D Co-ordinate Geometry Lecture-13 The ellipse Dated:-09.05.2020 PPT-05 UG (B.Sc., Part-1) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
The ellipse Directrix Definition:-An ellipse is the Locus of a point which Moves in a plane such That the ratio of its distance from the fixed point (called focus) and from the fixed line (called directrix) is always constant and less than unity. i.e. Where S is called focus and AZ is called directrix of the ellipse. B M Z X O F(focus) S(focus) N B a e = x PS PM = = tan 1( ) Cons t e eccentricity
Standard equation of an Ellipse a e Let S (ae,0) be the focus and be the eq. of the given directrix ZM of the ellipse,where e is the eccentricity. is the eccentricity. Let P(x,y) be any point on the Ellipse, then by the definition of the ellipse, From , PN OZ and PM AZ then ON x and PN y a Now PM NZ OZ OM x as OZ e = x B P(x,y) M Z A(a,0) X A (-a,0) O (0 1) e N S(ae,0) a e = B x PS PM = = .....(1) e PS ePM = = a e = = = = , ...(2)
Proof Continue From (1),We get ( ) ( ) 2 2 = = PS ePM PS PM 2 a e ( ) ( ) 2 2 + = 2 0 x ae y e x 2 + a ex + + = 2 2 2 2 2 2 x aex a e y e e aex + + + = 2 2 2 2 2 2 2 2 2 x aex a e y a e x ( ) y ( ) = 2 2 2 2 2 1 1 x e y a e 2 2 x a + = 1 ( ) 2 2 2 1 a e 2 2 x a y b + = 1 t n a S dard equation of the Horizont al ellipse 2 2 ( ) = 2 2 2 1 where b a e
