Minimum area bounded between two curves

Question

What is the minimum area bounded between the curve $ y = \dfrac{x^2}4 $ and $y=ax+9$, where $a$ is some real number.

I tried to use application of derivative but not able to reach a final solution.

Answer 1

The line $y=ax+9$ will always lie above the parabola $y=\frac{x^2}4$, intersecting when

$$ax+9=\frac{x^2}4\implies x^2-4ax-36=0$$

at two points, $x_1$ and $x_2$, both dependent on $a$. (It's given that $a\in\mathbb R$, so the line is never vertical, hence there are two points of intersection.)

The area between the curves is then a function of $a$,

$$\operatorname{Area}(a)=\int_{x_1(a)}^{x_2(a)}\left((ax+9)-\frac{x^2}4\right)\,\mathrm dx$$

which can be minimized by applying the fundamental theorem of calculus. Can you continue from here?