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.
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 $\sin \theta = d/(2a)$.