For what values of a and b is the line 2x+y=b tangent to the parabola y=ax^2 when x=2
When $\displaystyle x = 2, y = 4a$.
$\displaystyle y' = 2ax = 4a$
$\displaystyle y - 4a = y'(x - 2) = 4a(x - 2) \Rightarrow y - 4a = 4ax - 8a \Rightarrow y = 4ax - 4a$
Comparing $\displaystyle y = 4ax - 4a$ and $\displaystyle 2x + y = b$,
Writing it:$\displaystyle -4a = -4ax + y$ makes it easy.
$\displaystyle 2 = -4a = b \Rightarrow a = -\frac{1}{2} , b = 2$