Answer
When $a=-\frac{1}{2}$ and $b=2$, the requirement of the exercise is satisfied.
Work Step by Step
$$(l): 2x+y=b$$ $$(l): y=-2x+b$$
The slope of line $(l)$ is $-2$.
$$(S): y=f(x)=ax^2$$
- Derivative of $f(x)$: $$f'(x)=2ax$$
Line $(l)$ can only be tangent with parabola $(S)$ when $x=2$ in the case that $f'(2)$ equals the slope of line $(l)$.
In other words, $$f'(2)=-2$$ $$2a\times2=-2$$ $$4a=-2$$ $$a=-\frac{1}{2}$$
Therefore, the equation of parabola $(S)$ is $$(S):y=f(x)=-\frac{1}{2}x^2$$
- According to the equation of parabola $(S)$, when $x=2$, $y=-\frac{1}{2}\times2^2=-2$
So, point $A(2,-2)$ lies in parabola $(S)$.
However, since point $A$ is where tangent line $(l)$ touches parabola $(S)$, $A$ also lies in line $(l)$, which means $$2\times2-2=b$$ $$b=2$$
In conclusion, $a=-\frac{1}{2}$ and $b=2$ is the solution to this exercise.
