5. Given the family of curves

$$
y=(m-1) x^{2}+2 m x+4
$$

a) Prove that all the given curves pass through two fixed points $A$ and $B$. Determine these points.

b) Determine the curve from the given family that touches the $O x$ axis, and then determine the curve whose vertex is point B ( $B$ is the common point of the given curves, which does not lie on the $O x$ axis ).
Solution. a) Obviously, all curves of the family pass through the points $A(-2,0)$ and $B(0,4)$.

b) The $O x$ axis is tangent to the curve $y=(x+2)^{2}$, which is obtained for $m=2$, and the curve with the vertex at point $B$ is $y=-x^{2}+4$, which is obtained for $m=0$.