Descartes pointed out that the curves x^2=ay, x^2+y^2=ay+bx intersect in a point (x,y) such that x and y are the two mean proportionals between a and b. Show this.
Hello, sjenkins!
We must show that . form a geometric progression.Descartes pointed out that the curves: .
intersect in a point such that and are the two mean proportionals
between and . .Show this.
From [1], we have: .
Substitute into [2]: .
Square both sides: .
. .
Disregarding the trivial solution, we have: .
Substitute into [3]: .
The progression is: .
. . The common ratio is: .