A plane intersecting a double-napped cone is perpendicular to the central axis. Which conic section is formed?

Mathematics

2 answers:

kkurt [141]2 years ago

7 0

A) Circle

Correct on Edge 2021

Vitek1552 [10]2 years ago

7 0

Answer:

Step-by-step explanation:

Edge 2021

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Ipatiy [6.2K]

First notice that the triangle with sides $x,y,a$ and the triangle with sides $z,a+b,x$ are similar. This is true because the angle between sides $x,y$ in the smaller triangle is clearly $30^\circ$ , while the angle between sides $y,z$ in the larger triangle is clearly $60^\circ$ . So the triangles are similar with sides $x,y,a$ corresponding to $a+b,z,x$ , respectively.

Now both triangles are $30^\circ-60^\circ-90^\circ$ , which means there's a convenient ratio between its sides. If the length of the shortest leg is $\ell_1$ , then the length of the longer leg is $\ell_2=\sqrt3\ell_1$ and the hypotenuse has length $\ell_3=\sqrt{{\ell_1}^2+{\ell_2}^2}=2\ell_1$ .

Since $x$ is the shortest leg in the larger triangle, it follows that $a+b=2x$ , so $a+b=15=2x\implies x=\dfrac{15}2$