Question

If centre of circle is \(\large \bf (\pi,e)\),then how many rational points will lie on circle ?

Answer 1

\[
(x-\pi)^2+(y-e)^2=r^2
\]

Answer 2

correct,then

Answer 3

Technically speaking, I suppose \((0,0)\) could lie on the circle with the right radius.

Answer 4

ok

Answer 5

\[
x^2-2\pi x+\pi^2+y^2-2e y+e^2 = r^2
\]

Answer 6

then

Answer 7

I'm thinking you would analyze how the terms would be in terms of being rational or irrational.

Answer 8

yeah !!

Answer 9

and we know that
\[\large \bf rational \neq irrational\]

Answer 10

so answer would be `0`

Answer 11

Irrational times a rational is irrational.

Answer 12

correct

Answer 13

then what do you want to say ? @wio

Answer 14

doesn't it depend on whether the radius is rational or not ?

Answer 15

you have already got one rational point on the circle if you let the circle pass through a rational point to start with

Answer 16

i don't think radius is irrational

Answer 17

how ? ganehie

Answer 18

\[r^2 = (x-\pi)^2+(y-e)^2,~~~~r>0\] I think this makes more sense

Answer 19

let the circle pass through (0, 0)

Answer 20

\[
x^2+y^2= r^2 +2\pi x-\pi^2+2e y-e^2
\]It's possible that irrational numbers cancel themselves out.

Answer 21

ok.. @ganeshie8

Answer 22

no. @wio

Answer 23

We could let the radius equal: \[
\sqrt{\pi^2+e^2}
\]In this case: \[
(x-\pi)^2+(y-e)^2=\pi^2+e^2
\]And the point \((0,0)\) is on the circle.

Answer 24

lets see if we can find one more rational point

Answer 25

oh !! i see

Answer 26

so (0,0) is one rational point which lies on circle

Answer 27

correct ??

Answer 28

yes

Answer 29

if we pass the circle from point (1,1) then

Answer 30

I think you can calculate the distance between the centre to the point.

Answer 31

you don't know yet whether you can pass the same circle through both (0,0) and (1,1) or not

Answer 32

Giving you the radius

Answer 33

hmm any rational point is possible given an irrational radius

Answer 34

\[
r = \sqrt{(x_1-\pi)^2+(y_1-e)^2}
\]You can get any point \(x_1,y_1\) in the circle with the right radius.

Answer 35

Yes

Answer 36

if the point \(\large \bf (x_1,y_1)=(1,1)\),then

Answer 37

@wio

Answer 38

for definiteness, lets construct a circle that passes through (0, 0)

Answer 39

and see if we can find any other rational points on this circle

Answer 40

no,if we pass the circle through (0,0)

Answer 41

right ???

Answer 42

\[
r^2=(1-\pi)^2+(1-e)^2 = (x-\pi)^2+(y-e)^2
\]So \((1,1)\) can be on the circle.

Answer 43

it will lie on circle or not ??

Answer 44

(0,0) and (1,1) canot lie on the circle at the same time

Answer 45

correct !

Answer 46

but that observation is useless, it wont help us going forward

Answer 47

i know what i do to move forward as @wio tells me !
@ganeshie8

Answer 48

*told

Answer 49

idk what to do yet, im still thinking. what do you have in mind ?

Answer 50

now my question is this :-
\[\large \bf if~we~pass~circle~through~(1,1)~instead~of(0,0),~then\]

Answer 51

then nothign happens, its the same story :O

Answer 52

You can use the distance formula and check for yourself

Answer 53

@ganeshie8
if circle pass through (0,0),then
\[\large \bf (x- \pi)^2+(y-e)^2=r^2\]

Answer 54

you got one rational point on ur circle and you're stuck at the same place

Answer 55

plug (0,0),
\[\large \bf r^2=\pi^2+e^2\]

Answer 56

\[\large \bf \pi^2+e^2=r^2\]
\[\large \bf \pi^2+e^2=\pi^2+e^2\]
so we can say that (0,0) is rational point which lie on circle

Answer 57

@ganeshie8 and @wio