This is a very interesting analytic geometry math problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun time?! So anyway I figured out the answer and posted it below so you can see (This is a Q&A post). So without further ado, the question is:
Find the equation of the line through ($4,-1$) and whose segment intercepted by the axes has a length equal to $2 \sqrt{17}$ .
I also made a picture of the line segments and intersections ($4$ intercepts) in the diagram which you can see below, so make the question easier to understand.
There are 4 such lines, with the intercepted segment being in the first, third, or fourth quadrant (two of these), corresponding to intercepts where are:
$+x, +y$ (1st quadrant)
$-x, -y$ (3rd quadrant)
$+x, -y$ (4th quadrant) (2 of these)
(There isn't one with $-x, +y$)
Let the intercepts be $x,0$ and $0,y$
We must solve:
$x² + y²$ = 68 (square of distance of intercepts)
and
$\frac{y}{-x}$ = $\frac{-1-0}{4-x}$ $×$ (slope $y,0$ $\rightarrow$ x,0 = slope $4,-1$ $\rightarrow$ $x,0$ )
$\frac{y}{x}$ = $\frac{1}{4-x}$
$y = \frac{x}{4-x}$
$ \frac{x² + x²}{16-8x+x²} = 68$
$x² × (16-8x+x²) + x² = 68 ×(16 - 8x + x²)$
$16x² - 8x³ + x^4 + x² = 68×16 - 8×68 x + 68x²$
$ x^4 - 8x³ - 51x² + 544x - 1088 = 0$
From the graph and successive approximations, one finds roots at
$4.691885, -3.95189852042677e-005$
$3.526847, -7.87054668762721e-005$
$-8.218732475, -4.08132482334622e-005$
$8, 0$
Those are the $4$ $x$-intercepts of the lines.
The $8,0$ one is actually easy to find. If defines a line from $(0,-2) \rightarrow (4,-1) \rightarrow (8,0)$ so distance from $0,-2$ to $8,0$ is $2 \sqrt{17}$ by Pythagorean theorem.
For the others, the $y$-intercept is given by $y = \frac{x}{4-x}$
And the lines are defined by these points:
(blue) $0, -2 \rightarrow 8, 0$
(red) $4.691885,0 \rightarrow 0, -6.78130758724354$
(green) $3.526847, 0 \rightarrow 0, 7.45392505172746$
(turquoise) $-8.218732475, 0 \rightarrow 0, -0.672633801567867$
These points are all $\sqrt{68}$ apart.