Elliptical section of a right circular cone by HCR

Geometry Level 5

A right circular cone, with apex angle $\alpha =60^\circ$ , is thoroughly cut with a smooth plane inclined at an acute angle $\theta =60^\circ$ with the geometrical axis to generate an elliptical section. (As shown in the above diagram). Find the eccentricity of elliptical section generated.

The answer is 0.577350269.

2 solutions

Harish Chandra Rajpoot
May 15, 2018

In general, eccentricity ( $e$ ) of elliptical section generated is given by following formula

$\boxed{e=\frac{\cos\theta}{\cos\frac{\alpha}{2}}}$

$\therefore e=\frac{\cos60^\circ}{\cos30^\circ}=\frac{1/2}{\sqrt3/2}=1/\sqrt3=0.577350269$

Tanishq Varshney
May 12, 2015

Frankly typing , i just guessed it as $\frac{\cos 60}{\sin 60}=0.577$

@Harish Chandra Rajpoot Now I know what you mean by HCR Infinite series . " @Harish Chandra Rajpoot " infinite series. I just got a reseach paper of yours , its very interesting thank you for giving an idea of it and posting this problem.

Gaurav Jain - 6 years, 1 month ago

can you please explain how to solve this problem.

Rohan Chandra - 6 years, 1 month ago

Sorry for being late. This problem uses a simple result of a lengthy calculation by taking 3d polar co.ordinates about two axes. i.e. one in the X-Y plane taking around a circular arc as rcos(x) and another the projection of same about the this conic part in z-direction, major axis comes out /(2a=\frac { hsin(2\alpha ) }{ { sin }^{ 2 }(\theta )-{ sin }^{ 2 }(\alpha ) } /) and minor axis as /(2b=\frac { 2hsin(2\alpha ) }{ \sqrt { { sin }^{ 2 }(\theta )-{ sin }^{ 2 }(\alpha ) } } /)

Gaurav Jain - 6 years ago

Got a full solution?

Pi Han Goh - 6 years, 1 month ago

Elliptical section of a right circular cone by HCR

The answer is 0.577350269.

2 solutions

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