Shortest line segment that is cut off by the first quadrant and passes through a given point
Edit: this was posted before the question was edited to reflect the clarification. The following is the answer to a different problem:
If $b<a$, it's the line segment along $x=a$ with length $b$. Otherwise, it's the line segment along $y=b$ with length $a$ (if $a=b$, then they're the same length). You can see this by noting that if you fix the point at (a,b) but move the line, the distance will increase according to the Pythagorean theorem. So the answer is $\min(a,b)$.
Chris
Updated on July 31, 2022Comments

Chris 3 months
Let $a$ and $b$ be two positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point $(a,b)$.
I attempted to let the $y$intercept be $(0, Y_c)$ where the slope from there to $(a, b)$ is $(b  Y_c ) / a$ and $x$intercept be $(X_c, 0)$ and slope from there to $(a, b)$ is $(b / a  X_c)$. Since the two slopes are equal, they can be used to solve for $X_c$ in terms of $Y_c$ and $a, b$. Then use the distance formula to find distance $D$ between $x$intercept and $y$intercept points. Then differentiate $ D$ to find its max/min value.

Rory Daulton over 7 yearsWelcome to MathSE! However, this is not a homeworkanswering site. You need to give us more details, such as the attempts you have made and where you are stuck, as well as the level of math you want in the solution. You should also clarify what you are asking. Do you mean the shortest segment through $(a,b)$ with endpoints on the positive $x$ and $y$ axes?

Chris over 7 yearsCollege level calculus 1

Rory Daulton over 7 yearsGood: now the other questions I asked?

Chris over 7 yearsI attempted to let the yintercept be (0, Yc) where the slope from there to (a, b) is (b  Yc ) / a) and xintercept be (Xc, 0) and slope from there to (a, b) is (b / a  Xc). Since the two slopes are equal, they can be used to solve for Xc in terms of Yc and a, b. Then use the distance formula to find distance D between xintercept and yintercept points. Then differentiate D to find it's max/min value.
