Animated Solution for Mathematics - Straight Lines: If $(a,a^2)$ falls inside the angle made by the lines $y=\frac{x}{2}$, $x>0$ and $y=3x$, $x>0$, then $a$ belong to

Imagine you are standing at the origin of a coordinate plane, looking out into the first quadrant. You see two paths stretching out before you: one is a steep, aggressive climb defined by the line $y=3x$, and the other is a gentle, leisurely slope defined by $y=\frac{x}{2}$.

You are tasked with finding a special kind of traveler—a point $P(a,a^2)$—that must stay within the 'wedge' formed by these two lines. This is not just an algebra problem; it is a story of confinement and freedom.

Our traveler, $P(a,a^2)$, is not moving randomly. Its coordinates are locked in a dance: the $y$-coordinate is always the square of the $x$-coordinate.

This means our point is permanently tethered to the curve of the parabola $y=x^2$. As $a$ changes, the point slides along this elegant, curved path.

To be 'inside' the angle means that for any given $x$-coordinate $a$, the $y$-coordinate of our point must be greater than the lower boundary and less than the upper boundary. Mathematically, this translates to the following compound inequality:

This inequality is the gatekeeper. It defines the exact range of $a$ that allows our point to exist within the wedge. We must solve this in two parts.

First, we ensure the point stays below the steep line: $a^2<3a$. Rearranging this, we get $a^2-3a<0$, or $a(a-3)<0$. Since we know $a>0$, we can divide by $a$ to find $a<3$.

Second, we ensure the point stays above the gentle line: $\frac{a}{2}<a^2$. Rearranging this gives $a^2-\frac{a}{2}>0$, or $a(a-\frac{1}{2})>0$. Again, dividing by $a$ (because $a>0$), we find $a>\frac{1}{2}$.

We have arrived at the climax of our journey. We have two conditions that must be satisfied simultaneously: $a<3$ and $a>\frac{1}{2}$.

When we combine these, we find that $a$ must live in the open interval:

This is the 'sweet spot' where our parabolic traveler is safely nestled between the two lines. It is a perfect, elegant solution that balances the rigidity of lines with the fluidity of a parabola.