TITLE: Obtaining the equation of an ellipse with only information about the diameter and an angle
QUESTION [1 upvotes]: I am dealing with the following word problem:
A spotlight throws a beam of light that is 25cm in diameter. If the beam hits the stage floor at an angle of $60 ^\circ$ with the horizontal, find an equation for the elliptical pool of light on the stage floor 
(from Gersting, Technical Calculus with Analytical Geometry)
I have started off this problem with the ellipse equation  $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ and set the axes at the centre of the ellipse.
I divide the diameter by 2 to give me the length of the semi-minor axis and the location of the y intercept. I then substitute this into the equation to get $\frac{x^2}{a^2} + \frac{y^2}{12.5^2}=1$. Now I don't know what to do. It seems like the angle given in the information is important, but I can't see how. Isn't the angle that the spotlight hits the floor on a different plane to the ellipse?
Thanks in advance.

REPLY [1 votes]: This rather poorly worded (by the author of the text, not the poster) question needs to state that the "spotlight" is assumed to be a cylinder of diameter $25$ cm, and that the ellipse in question is the intersection of said cylinder by a plane that forms an angle of $60^\circ$ with the cylinder's axis.  In that case, the minor axis of the ellipse clearly remains $25$ cm--the diameter of the cylinder.
The major axis will satisfy the relationship $$2a = \frac{d}{\sin \theta},$$ where $d = 25$ is the diameter of the cylinder and $\theta = 60^\circ$ is the aforementioned angle.  So we have $a = 25/\sqrt{3}$.  To see why this relationship is true, consider the intersection of the cylinder with a plane perpendicular to the cylinder's axis and incident to either one of the two vertices of the ellipse at the major axis.  This creates a circle with diameter $d$, and inside the cylinder, we then have a right triangle with one leg of length $d$ and the hypotenuse $2a$.  The angle subtending the leg of length $d$ is $\theta$, hence $\sin \theta = d/(2a)$.
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