The chord of circle #x^2 + y^2 = 25# connecting #A(0,5)# and #B(4,3)# is the base of an equilateral triangle with the third vertex #C# in the first quadrant. What is the smallest possible #x#-coordinate of #C#?
5 Answers
Smallest possible x-coordinate of
Explanation:
given a circle
slope of
let
As
as
slope of
Let co-ordinates of
Smallest possible x-co-ordinate of
Explanation:
solution 2 :
Given a circle
let
as
slope of
Let co-ordinates of
Hence, smallest possible x-co-ordinate of
Explanation:
Let the third vertex of the equilateral
We have,
Using the distance formula,
Also
Clearly,
Thus,
Respected CW has readily obtained!
Explanation:
We're oddly given a circle in this problem and awkward wording asking for the smallest possible x coordinate.
I'm not sure what the circle has to do with anything. We could list equations endlessly for circles containing these two points. Forget about the circle.
Given one side, there will be two equilateral triangles with that as a side. Let's rewrite the problem:
Find all possible third vertices
There are a few different ways to do this. I'd lean toward complex numbers, but I probably should try to keep this simpler than that.
Let's review. The altitude
We just need to go
The midpoint D of AB is
The direction vector from A to B is
For the perpendicular direction vector we swap and negate one:
Now schematically what we're doing is
We have
That's the general solution; let's apply it to
Uh, the one with the least x coordinate is
Check:
We check the squared distance to each point
Explanation:
I did a geometric construction turned into algebra via the Cartesian plane in the last answer. Let's try something a bit different here.
Given
Two equations, two unknowns. First one first.
That must be the equation for the perpendicular bisector of AB.
The Shakespeare Quadratic Formula (
The smallest root is