Semi vertical angle of Right circular cone

Question

Show that the semi vertical angle of the right circular cone $$4(x^2+y^2)-9z^2=0$$ is $$\arctan\left(\frac{3}{2}\right)$$

I took a cone whose vertex is at zero and axis is z axis , then I found the equation of cone and compared it with the given equation and had the answer.

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Raffaele · Accepted Answer · 2017-09-20T09:58:58.740

0

Substitute $y=0$ and get $4x^2-9z^2=0$

factor $(2x-3z)(2x+3z)=0$ which gives the equation of the two lines $x=\frac{3}{2}\,z$ and $x=-\frac{3}{2}\,z$.

The semivertical angle is the angle formed by one of this lines in the plane $y=0$ with the simmetry axis of the cone. The slope $\frac{3}{2}$ is the $\tan$ of the requested angle $\alpha$, so we have $\alpha=\arctan\left(\frac{3}{2}\right)$

edited Sep 20 '17 at 09:58

answered Sep 20 '17 at 09:56

Raffaele

26,371

Why will I substitute y=0? – debapriya basu Sep 20 '17 at 09:58
Because I want the intersections with the plane $y=0$ which are two lines of the cone which are $180°$ one from the other – Raffaele Sep 20 '17 at 10:00

Semi vertical angle of Right circular cone

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