1. Given the quadratic polynomial $(k+1) x^{2}-2 k x+k-1$, where $k$ is a real parameter.

a) Determine the geometric locus of the vertices of all parabolas defined by the equation

$$
y=(k+1) x^{2}-2 k x+k-1
$$

b) Do all these parabolas have a common point?

c) Prove that only one solution of the equation

$$
(k+1) x^{2}-2 k x+k-1=0
$$

is variable and illustrate it graphically.

d) Determine for which natural number $k$ the variable solution of equation (1) is a periodic decimal number with one digit in the period.
Solution. a) The vertex of the parabola

$$
y=(k+1) x^{2}-2 k x+k-1, \text { where } k \neq-1
$$

has coordinates

$$
x=1-\frac{1}{k+1}, y=-\frac{1}{k+1}
$$

The geometric locus of the vertices of these parabolas is the set $\{(x, y) \mid y=x-1, x \neq 1\}$.

b) Each parabola of the given family

![](https://cdn.mathpix.com/cropped/2024_06_05_faf575a75202fd173f3cg-01.jpg?height=360&width=521&top_left_y=995&top_left_x=958)
contains the point $(1,0)$, as shown in the diagram on the right.

c) The solutions to equation (1) are $x_{1}=1$ and $x_{2}=\frac{k-1}{k+1}$.

d) The required numbers are $2, 5, 8$ and $17$.