Calculating Lamp Location Above x-Axis: Elliptical Shadow Problem

[boarding4lif3]

Homework Statement

A lamp is located three units to the right of the y-axis and a shadow is created by the elliptical region x^2 + 4y^2 = 5. If the point (-5,0) is on the edge of the shadow, how far above the x-axis is the lamp located?

2. The attempt at a solution
Ive calculated the derivative with respect to X to be -x/4y = y'. But when i try to find the slope, I am stuck with an undefined when i try to use the point (-5,0)

Any help would be appriciated. Thanks.

 

[HallsofIvy]

Well, the ellipse itself does not include the point (-5, 0) so there is no tangent to the ellipse there. What you want is a line through (-5, 0) that is tangent to the ellipse at some other point.

Let $(x_0,y_0)$ be the point of tangency. The line through (-5, 0) and $(x_0, y_0)$ has slope $(y_0- 0)/(x_0-(-5))= y_0/(x_0+ 5)$.

Yes, the derivative is given by $y'= -x/(4y)$ and at $(x_0, y_0)$ that is $-x_0/(4y_0)$.

So you want to find $(x_0, y_0)$ satisfying both $y_0/(x_0+ 5)= -x_0/(4y_0)$ and the equation of the ellipse, $x_0^2+ 4y_0^2= 5$