Solution assignment 09 Root equations

Assignment 9

For which value(s) of $c$ is the line:

$l: y=cx$

tangent to the graph of:

$k: y=\sqrt{x-1}$

Solution

First we try to find possible intersection points of $l$ and $k$ . Next we try to choose $c$ such that the intersection points coincide (or graphs have 'one' intersection point).
We find these intersection points of $l$ and $k$ from:

$cx=\sqrt{x-1}$

After squaring we find:

$c^2x^2=x-1$

$c^2x^2-x+1=0$

The solutions of this quadratic equations indicate the intersection points of $l$ and $k$ . The discriminant determines whether there are intersection points and if so, how many: 0, 1 or 2. We want that the graphs of $l$ and $k$ are tangent and thus $D=0$ :