Example 9. At what angles do the curves $y=x^{2}$, $y^{2}=x$ intersect?
Solution. The angle between two curves at their point of intersection is called the angle between the tangents to the curves at this point.

Let's find the points of intersection of the given lines. Solving the system of their equations

$$
\left.\begin{array}{l}
y=x^{2} \\
y^{2}=x,
\end{array}\right\}
$$

we get two points of intersection: $O(0,0), \quad M(1,1)$.

The equations of the tangents to the curves at any point will be, respectively:

$$
\begin{aligned}
& Y-y=2 x(X-x) \\
& Y-y=\frac{1}{2 \sqrt{x}}(X-x)
\end{aligned}
$$

in particular, at point $O(0,0)$

$$
Y=0, X=0
$$

and at point $M(1,1)$

$$
Y-1=2(x-1), Y-1=\frac{1}{2}(X-1)
$$

As can be seen from the equations, the tangents to the curves at point $O(0,0)$ coincide with the coordinate axes, so the angle between the curves is $90^{\circ}$.

At point $M(1,1)$, the angle between the lines is determined by the formula

$$
\operatorname{tg} \varphi=\frac{k_{2}-k_{1}}{1+k_{1} \cdot k_{2}}=\frac{2-\frac{1}{2}}{1+2 \cdot \frac{1}{2}}=\frac{3}{4}, \varphi=\operatorname{arctg} \frac{3}{4}
$$

where $k_{1}=2, k_{2}=\frac{1}{2}$ are the slopes of the tangents.