Is it possible to draw a circle with any pairs of points above x axis such that the circle is tangent to x axis?
#Hardest geometry problem I have ever faced?!
carmine valley
gray plume
Is it possible to draw a circle with any pairs of points above x axis such that ...
(x-l)²+(y-r)²=r²
l just shifts it left or right depending on where you want it centred
And r should be positive if you want it above the x-axis
gray plume
Like what is l and r?
r is the radius of the circle, l is just any variable that shifts it in the x direction
gray plume
Wait so you want to find the equation of a circle that passes through 2 points and is tangent
carmine valley
To the x axis
gray plume
Ok one second
carmine valley
This circle includes 0,2 and 1,1 and is tangent to x axis
Given any two points above x axis, would it be possible then?
@gray plume
Bro you there?
gray plume
yes im just trying to make a general formula
carmine valley
Ok thanks
I wonder
Does it go like this?
The two points can form a chord of the circle, and we know the perpendicular bisector passes through the centre, find a point on the bisector line such that the distance to the two points and the tangent is the same
gray plume
gray plume
equation isnt looking too good so far
carmine valley
Ah looks like there are two cases
gray plume
solving this gives the x coordinate of the centre of the circle
carmine valley
Fu….
gray plume
ill just send you the graph when im done
carmine valley
You mean this?
There seems to be two cases: when one of the coordinates of the points match (like either same x or y or both) and when they are different
gray plume
another thing to do is see that the circle will be tangent at (l,0) where l is the x value for the centre of the circle
carmine valley
Well yeah
gray plume
then you can use one of the points to get another chord bisecting it and where they cross will be the centre
carmine valley
The y coordinate of the centre is equal to the radius
gray plume
yes
carmine valley
then you can use one of the points to get another chord bisecting it and where t...
Isn’t it self inflicted
gray plume
but right now it only seems to be for the case with the smaller circle
gray plume
Isn’t it self inflicted
i dont know what that means
carmine valley
Idk how to describe
But like to find another point you need to know the eq of circle right?
gray plume
you said we already know 2 points
the 3rd which is on the line y=0 means we have 3 points
carmine valley
Oh ok
gray plume
which is the minimum amount of points to describe a circle
carmine valley
I see
gray plume
this is what ive got so far, (x₁, y₁) and (x₂, y₂) are two points that are given, and if you have a slider for you can manually make it pass through the two points on both sides
ill continue later, but if you want to try and continue earlier then you just need to find a way to make l dependend the two points
delicate magnet