Crazy Elliptical shadow problem

[metalmagik]

Homework Statement

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?

Homework Equations

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

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.

 

[HallsofIvy]

Homework Statement

The figure shows a lamp located three units to the right of the y-axis and a shadow 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?

Homework Equations

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

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.

The "light ray" from from (3, y_L) 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 let's you solve for both x and y_L. It is the latter you want.

 

[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.