4. Calculate the parameter $a$ so that the parabola with the equation $y=(a-1) x^{2}+a x-1$ and the line with the equation $y=x-2$ have only one common point. Write the coordinates of this point.
4. We write the equation $(a-1) x^{2}+a x-1=x-2$ and rearrange it to $(a-1) x^{2}+(a-1) x+1=0$. The equation must have exactly one solution, so we consider the condition $D=0$. We get the equation $a^{2}-6 a+5=0$. The solutions to this equation are $a_{1}=1$ and $a_{2}=5$. We exclude the first solution, as for $a=1$ the equation $y=(a-1) x^{2}+a x-1$ is not a parabola. We calculate the coordinates of the intersection point $M\left(-\frac{1}{2},-\frac{5}{2}\right)$.

Zapis enačbe $(a-1) x^{2}+a x-1=x-2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=51&width=1639&top_left_y=563&top_left_x=208)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=51&width=1642&top_left_y=614&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=49&width=1642&top_left_y=661&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=57&width=1631&top_left_y=705&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=66&width=1642&top_left_y=755&top_left_x=207)