Understanding the solution to a problem on Lagrange Multipliers

Question

This is a problem from Riley-Hobson "Mathematical Methods For Physics And Engineering".

QUESTION: Two horizontal corridors, $0\le x \le a$ with $y\ge 0$ and $0\le y \le b$ with $x\ge 0$, meet at right angles. Find the length of the longest ladder that can be moved around the corner of the corridor. Use Lagrange Multipliers to solve the problem.

I checked the following questions.

1. Longest pipe that fits around a corner.

2. Intuitive explanation for formula of maximum length of a pipe moving around a corner?

that are almost similar.

However I found only one solution which had an approach of Lagrange multipliers.

In that solution, the author writes as follows:

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For example, if we rephrase the question as

Find the length of the shortest line segment from $(x,0)$ to $(0,y)$ that passes through $(a,b)$.

then with a little coordinate geometry we can rephrase it again as

Minimize $(x^2+y^2)^{1/2}$ subject to the constraint $\frac ax + \frac by = 1$.

Using the method of Lagrange multipliers from multivariable calculus on this version of the problem yields quickly that the optimum has $(x^3,y^3)$ proportional to $(a,b)$.

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Can someone explain how he first makes that rephrasal and then how he derives the constraints and makes that conclusion ?

P.S. The author is not on MathsSE anymore, hence I cannot ask him for clarification.

Answer 1

To do the conversion, $(x^2+y^2)^{1/2}$ is simply the length by Pythagoras.

The line which passes through $(x,0)$ and $(0,y)$ is $xy=Xy+xY$ (variables are $X$ and $Y$), and this also passes through $(a,b)$ iff $xy=ay+xb$ or $1=\frac ax+\frac by$, which is the constraint.

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Then taking $$L=(x^2+y^2)^{1/2}+\lambda \left (\frac ax+\frac by-1\right)$$ and using $d=(x^2+y^2)^{1/2}$ for the length of the ladder, we want the partial derivatives of $L$ to be zero i.e. $$L_x=\frac xd-\frac {\lambda a}{x^2}=0$$and $$L_y=\frac yd-\frac {\lambda b}{y^2}=0$$ whence the solution has $$x^3=\lambda ad; y^3=\lambda bd$$ and dividing one by the other gives $$\frac {x^3}{y^3}=\frac ab$$

Answer 2

It is quite obvious that the critical position for the ladder is the one where the ladder touches $(a,b)$ and the endpoints are at $(x,0)$ and $(0,y)$ (along the outer boundary of the corridors). The reason is the following: To move the ladder around the corner, you can for example keep a point of the ladder at $(a,b)$ then you move one of the endpoints along $(x,0)$. As long as the other endpoints is at a position $(x_2, y)$ with $x_2\leq 0$ the ladder "fits". The longest length of the ladder is thus the shortest distance between $(x,0)$ and $(0,y)$ while passing through $(a,b)$.

The length of the ladder is given by $(x^2+y^2)^{1/2}$. So we only need to determine the condition that the line passing through $(x,0)$ and $(0,y)$ also passes through $(a,b)$. We can write the expression $$ \frac{y_l}{y} + \frac{x_l}{x} =1 \tag{1}$$ for the points $(x_l,y_l)$ on the line segment of the critical position of the ladder. Demanding that $x_l=a$, $y_l=b$ fulfils (1) leads to the result quoted.