# Homework Help: Crazy Elliptical shadow problem

1. Oct 5, 2008

### metalmagik

1. The problem statement, all variables and given/known data
The figure shows a lamp located three units to the right of the y-axis and a show created by the elliptical region x2+4y2$$\leq$$5. If the point (-5,0) is on the edge of the shadow, how far above the x-axis is the lamp located?

2. Relevant equations
Implicit Differentiation has to fit in here, since the chapter is on Implicit Differentiation, just not sure how.

3. The attempt at a solution
I figure the tangent line has to be at (-1,1) to be able to hit the lamp three units to the right...but I really just am not sure how to really start this problem...very confusing, can someone give me a starting point? Thank you very much for any help.

2. Oct 5, 2008

### HallsofIvy

The "light ray" from from (3, yL) to (-5, 0) that forms the edge of the shadow must be tangent to the ellipse (do you see why? If not draw a picture!). Let (x, y) be the point on the ellipse at which that line is tangent to the ellipse. Since it is a point on the ellipse, x2+4y2$$\leq$$5 and you can use implicit differentiation to find the slope of the tangent line at that point as a function of y. That, together with the fact that (x,y) must be on the tangent line lets you solve for both x and yL. It is the latter you want.

3. Oct 5, 2008

### metalmagik

I differentiated x2+4y2$$\leq$$5 and got like dy/dx $$\leq$$ -2x/8y and then I did not know how to continue. I looked up the problem online and found another way to do it that I am not sure if its right, the person who did this solution said that the shadow area had to be translated so he did like (x+3),(y+a) or something and plugged that into the original equation, solved through and got 1/2 as the answer. Is this correct?? Help please I am having a lot of trouble understanding this problem. Thank you.