Question

can it be proved that the sum of the distances from any point on the ellipse to the foci of that ellipse is constant for all the points lying on the ellipse ?

Answer 1

proved? dunno; but that is the definition of an ellipse.

Answer 2

i am confused whether this is the definition of the ellipse or the one which says "ellipse is the locus of points whose ratio of the distance from a point (focus) and from a line (directrix) is constant (ecentricity) and is less than 1"

Answer 3

plz tell me which is one of the two is the definition of ellipse...

Answer 4

an ellipse has a constant sum of the distance between all points and the foci

Answer 5

this is the definition of ellipse

Answer 6

lets take a general pt on the ellipse
acost, bsint
now one focus of the ellipse is ae, 0 and the other is -ae,0

Answer 7

@him1618 i hav given two definitions above. plz tell me which one of the two is the standard definition of ellipse

Answer 8

now evaluate the distances from both the foci of this general pt
add them up...itll come out to be 2a

Answer 9

The ellipse is a conic which is the locus of all points whose distance from a fixed point called a focus , bears a constant ratio to its distance from a fixed line called a directrix. The ratio is less than 1, called the eccentricity

Answer 10

Also,
the ellipse is the locus of a point whose sum of distances from two fixed points is constant

Answer 11

so these are the two definitions

Answer 12

u mean one can be derived from another one?

Answer 13

yes

Answer 14

have u ever done it?
plz show me how to do it. (take both cases. i.e take first defn as fact then prove the other one and vice versa)

Answer 15

so ive already told u the method to find the second condition
if u derive the equation of an ellipse from its standard mathematical definition (the first one)..ull find that any general pt on an ellipse can be shown in parameter t as acost, bsint

Answer 16

that is extremely long to do....especially on a computer

Answer 17

b ut see

(x-h)^2 + (y-k)^2 = e^2[(ax+by +c)/ (a+b)]^2

Answer 18

ok then. thanx for ur time. tc

Answer 19

w8

Answer 20

now u dont go about expanding this....cz its one helluva calculation

Answer 21

so this is ur ellipse

Answer 22

now for simplicity sake, well take a standard ellipse centred at the origin, with length of major axis 2a, and minor axis 2b

which is
(x/a)^2 + (y/b)^2 = 1

Answer 23

u get that?

Answer 24

ya. thats quite clear. i appreciate:)

Answer 25

but u proved the eqn of ellipse and not what i wanted u to...

Answer 26

so if u observe carefully, u can represent this ellipse as
x=acost
y=bsint
now well see that a general point on the ellipse is (acost, bsint)

Answer 27

now if u remember the two foci of this ellipse were talking about are ae, 0 and -ae,0

Answer 28

be patient.....now if u evaluate the distances of this general point from these 2 foci and sum them up, the sum comes to 2a

Answer 29

get it?

Answer 30

i m not sure, i think i m getting headache. i'll look it again , later and then accordingly reply u... u have got patience. i m impressed.

Answer 31

keep in touch..dont get stressed its quite easy