2. Through the vertex of a right circular cone, two planes pass. One of them is inclined to the base of the cone at an angle $\alpha$ and intersects it along a chord of length $a$. The other is inclined to the base at an angle $\beta, (\beta \neq \alpha)$ and intersects it along a chord of length $b$. Determine the volume of the cone.
2. Let $|M N|=a, P$ be the midpoint of $\overline{M N}, \alpha=\angle K P S$. Then,

![](https://cdn.mathpix.com/cropped/2024_06_03_843fcc95a1719fe2319dg-09.jpg?height=497&width=426&top_left_y=1399&top_left_x=678)

$h=|P S| \tan \alpha, \quad|P S|=\sqrt{r^{2}-\left(\frac{a}{2}\right)^{2}} \Rightarrow h=\sqrt{r^{2}-\left(\frac{a}{2}\right)^{2}} \tan \alpha$

Similarly, $h=\sqrt{r^{2}-\left(\frac{b}{2}\right)^{2}} \tan \beta$.

5 points

From (1) and (2) we get

$$
r^{2}=\frac{\left(\frac{a}{2}\right)^{2} \tan ^{2} \alpha-\left(\frac{b}{2}\right)^{2} \tan ^{2} \beta}{\tan ^{2} \alpha-\tan ^{2} \beta}
$$

10 points

From (1) we have

$$
h=\sqrt{\frac{\left(\frac{a}{2}\right)^{2}-\left(\frac{b}{2}\right)^{2}}{\tan ^{2} \alpha-\tan ^{2} \beta}} \tan \alpha \tan \beta
$$

Now the volume of the cone is

$$
V=\frac{1}{3} r^{2} \pi h=\frac{\pi}{24} \tan \alpha \tan \beta \cdot \frac{a^{2} \tan ^{2} \alpha-b^{2} \tan ^{2} \beta}{\tan ^{2} \alpha-\tan ^{2} \beta} \cdot \sqrt{\frac{a^{2}-b^{2}}{\tan ^{2} \alpha-\tan ^{2} \beta}}
$$