1. ## Apps of differentiation

I don't have a problem solving these problems but i cant seem to find the equation(s) of this problem. A right triangle is formed in the first quadrant by the x and y axes and a line through the point (1,2). a0Write the length L of the hypotenuse as a function of x.

the second part asks for me to find the minimum length but i can do that on my own. I can't find the equations for it. I believe the primary is L^2= x^2+y^2 but i cant determine the secondary one to sub in for y. I appreciate any help.

2. Hello, jarny!

A right triangle is formed in the first quadrant by the x and y axes
and a line through the point (1,2).
a) Write the length L of the hypotenuse as a function of x.
Code:
        |
(0,b)*
| *
|   * (1,2)
|     o
|       *
|         *
- - + - - - - - * - -
|         (a,0)

The line with intercepts $(a,0),\0,b)" alt="(a,0),\0,b)" /> has the equation: . $\frac{x}{a} + \frac{y}{b} \:=\:1$

Since $(1,\,2)$ is on this line: . $\frac{1}{a} + \frac{2}{b} \:=\:1\quad\Rightarrow\quad b \:=\:\frac{2a}{a-1}\;\;{\color{blue}[1]}$

The length of the hypotenuse is given by: . $L^2 \:=\:a^2+b^2\;\;{\color{blue}[2]}$

Substitute [1] into [2]: . $L^2 \;=\;a^2 + \left(\frac{2a}{a-1}\right)^2 \;=\;\frac{a^4 - 2a^3 + 5a^2}{(a-1)^2}$

Therefore: . $L \;=\;\frac{\sqrt{x^4-2x^2 + 5x^2}}{x-1}$