Question

The equation of circle which passes through (1,0),(0,1) & has its Radius as smal as possible is?

Answer 1

There's only one unique circle passing through (1,0) and (0,1) right?

Answer 2

No.

Answer 3

i dont think so

Answer 4

Thanks for clearing my doubts. That's what I was trying out

Answer 5

May I have a counterexample BTW?

Answer 6

radius is your choice

Answer 7

Thanks.

Answer 8

Two circles that intersect, right. =)

Answer 9

the smallest circle is the one that has the line connecting the two points as the diameter.

Answer 10

wait what did you say :?

Answer 11

and why are there 2 circles

Answer 12

Radius of \(1\)?

Answer 13

That's just the unit circle.

Answer 14

all other circles have this line as their chord which means that their diameters are longer that this line. An extreme example:

Answer 15

If you draw a line segment, you'd get a segment with length \(\sqrt 2\). That is the smallest segment (and the only) possible. Now you get a triangle. Draw a circle around that triangle and that would be \(x^2 + y^2 = 1\).

Answer 16

why all this?

Answer 17

\[r^2 = h^2 + (k - 1)^2 = (h - 1)^2 + k^2\]

Answer 18

I just want to know why should they be diameterically opp..to satisfy the eq.nothing else :/

Answer 19

\[h^2 + k^2 -2k+1=h^2 - 2h + 1 + k^2\]\[-2h + 1 = -2k +1 \iff h =k\]So \((0,0)\)?

Answer 20

\((0,0)\) is the center.

Answer 21

why u no tell jitna pucha :(

Answer 22

1 answer

Answer 23

\[x^2 + y^2 = 1\]

Answer 24

Pakka yahi hai answer.

Answer 25

lol no way

Answer 26

Why this Kolaveri

Answer 27

the concept is pretty straightforward. Look at this diagram
The line you see is the line connecting (1,0) and (0,1)
And i've drawn two circles that pass though the two points.
The line drawn is a chord for the bigger circle which means that it's diameter is bigger that the line, ok?
But look at the other circle which has the line as the diameter. It's obviously smalller than the other circle but also smaller than any other circle that can be drawn though those two points (again because chords are smaller than diameters)
So, the circle you're looking for is the circle that has that line for it's diameter. Simple.

Answer 28

not so clear but yeah somewhat..

Answer 29

another way to think is that you're looking for the smallest right triangle \(ABC\) where \(\angle B\) is the right angle as well as the center of the circle and the circle passes through \(A\) and \(C\).

Answer 30

how can (1,0) and (0,1) be diameterically opp but :O

Answer 31

I didn't say they were diametrically opposite =/

Answer 32

:S

Answer 33

but i did.

Answer 34

they have to be.

Answer 35

They are diametrically perpendicular...

Answer 36

how on eart h:/

Answer 37

No parth. I don't even know what that means...
But i'll draw on paper and show you what I mean.