<s>[INST] Finding the slope of line intersecting the parabola A line $y=mx+c$ intersects the parabola $y=x^2$ at points $A$ and $B$. The line $AB$ intersects the $y$-axis at point $P$. If $AP−BP=1$, then find $m^2$. where $m > 0$.
so far I know $x^2−mx−c=0,$ and $P=(0,c)$.
$x = \frac{m \pm \sqrt{m^2 + 4c}}{2}$
$A_x = \frac{m + \sqrt{m^2 + 4c}}{2}$, $B_x = \frac{m - \sqrt{m^2 + 4c}}{2} $
$A_y = \frac{m^2 + m\sqrt{m^2 + 4c}}{2} + c$, $B_y = \frac{m^2 - m\sqrt{m^2 + 4c}}{2} + c$
using distance formula(not showing all steps)
$AP = \frac{m + \sqrt{m^2 + 4c}}{2}(\sqrt{m^2 + 1}) $
$BP = \frac{m - \sqrt{m^2 + 4c}}{2}(\sqrt{m^2 + 1}) $
$AP - BP = 1$
$(\sqrt{m^2 + 4c})(\sqrt{m^2 + 1}) = 1$
$m^4 + m^2(4c + 1) + 4c - 1 = 0$
well I could manipulate this into quadratic but that doesn't really help me with coefficient with c.
 [/INST] There is a much simpler solution. At the intersection of $y = mx + c$ and $y = x^2$,
$$x^2 - mx - c = 0$$
We are given $|AP - PB| = 1$ and I am taking the case where $P$ is interior to segment $AB$. Just for completeness sakes, if $P$ is exterior to segment $AB$, then we still have $|AP - PB| = 1$ if we consider $AP$ and $PB$ as signed distances or we can say $|AP| + |PB| = 1$. Coming back to the case where $P$ is interior to $AB$, if the roots are $x = \alpha, \beta$,
$ |\alpha + \beta| = |m|$.
But as $\alpha$ and $\beta$ have opposite signs and they are x-coordinates of points $A$ and $B$, $~|\alpha + \beta|~$ is the absolute difference of the horizontal projections of $AP$ and $PB$, which can also be written as $AP |\cos \theta|$ and $PB |\cos \theta|$, where $\tan \theta = m$ is the slope of the line.
$$ |\cos\theta| = \frac{1}{\sqrt{1+m^2}}$$
$$|AP - PB| \cdot |\cos \theta| = |\alpha + \beta| = |m|$$
As $|AP - PB| = 1$,
$$m \sqrt{m^2 + 1} = 1 \implies (m^2 + \frac 12)^2 = \frac 54$$
We get two real solutions for $m$,
$$m = \pm \sqrt{\frac{\sqrt 5 - 1}{2}}$$
You can draw a few lines with equation $$y = \pm \sqrt{\frac{\sqrt 5 - 1}{2}} ~x + c$$ for different values of $c$ with $c \gt 0$ to confirm $|AP - PB| = 1$
As a side note, for values of $c \lt 0$ as long as the line intersects the parabola at two points, $|AP| + |PB| = 1$
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