Question

What are the coordinates of the points a and b and minimize the length of the hypotenuse of a right triangle that is formed in the first quadrant by the x-axis, the y-axis, and a line through the point (1,2) where point a is at (0,y) and point b is at (x,0)?

Answer 1

`(x_0,y_0) = (1,2)`

Suppose that the base point along the Y-axis is some scaled factor, `s`, of `y_0` beyond `y_0\`

That is the point on the Y-axis #(0,y_a)# is at
`(0, y_0 + s*(y_0))`
or `(0,2 + 2s)` in our particular case

The point on the X-axis (x_b, 0) must satisfy the equivalent slope ratios:

`x_b / (2+2s) = 1/(2s)`

`rarr` `x_b = 1 + 1/s`

The square of the ladder length based on the scaling factor can be expressed as
`L^2(s) = (x_b)^2 + (y_a)^2`
`= (1+1/s)^2 + (2+2s)^2`

which simplifies to
`L^2(s) = s^(-2) + 2s^(-1) + 5 + 8s + 4s^2`

We want to minimize `L(s)`
Since `L(s)` is required to go through `(2,1)`, `L(s)` must be greater than `1`
and we can simplify our effort with the same results by minimizing `L^2(s)`

`(d (L^2(s))) /(ds)`
`= ((-2)s^(-3) + (-2)s^(-2) + (8 + 8s)`
`= (-2)/s^3 + (-2)/s^2 + 8s + 8`

Setting the derivative to zero to find the minimum length:
`(d (L^2(s))) /(ds) = 0` gives
`(-2)/s^3 + (-2)/s^2 + 8s + 8 = 0`

multiplying by `(s^3)` [valid since s can not be `0`]
`-2 - 2s + 8s^4 + 8s^3 = 0`

factoring out `(s+1)` [again valid since `s` can not be `-1`]
`(-2)(s+1) + (s+1)(8s^3)= 0`
`(-2) + 8s^3 = 0`
`8s^3 = 2`
`s^3 = 2/(2^3) = 1/(2^2)`
`s = 1/4^(1/3)`

Recalling that
`y_a = 2+2s` and `x_b = 1 + 1/s`

we have
`y_a = 2 + 2/4^(1/3)`
`= 3.26` (approx.)
and
`x_b = 1 + 4^(1/3)`
`= 2.59` (approx.)

No sense stopping now.
From our earlier formula
`L^2(s) = s^(-2) + 2s^(-1) + 5 + 8s + 4s^2`

`L^2(s=1/4^(1/3)) = (1/4^(1/3))^(-2) + 2(1/4^(1/3))^(-1) + 5 + 8(1/4^(1/3)) + 4(1/4^(1/3))^2`

`L^2(s=1/4^(1/3)) = 17.32` (approx.)

`L(s=1/4^(1/3)) = 4.16` (approx.)

Congratulations to anyone who made it this far!
I hope there is a simpler method, but I couldn't find it.
I also hope that this is somewhere approaching correct and I didn't mess up.