Possibly flawed calculus homework question (tangent to ellipse) I have an online homework program called Web Assign for my calculus course. It has given me this problem:
Find equations of both the tangent lines to the ellipse x^2 + 9y^2 = 81 that pass through the point (27, 3).
First, unless I'm much mistaken, that point does not lie on the ellipse at all. Something is fishy already. What's more, unless I am mistaken yet again, ellipses always have one tangent line to every point. Regardless, I went ahead and did the problem using implicit differentiation.
2x + 18yy' = 0
18yy' = -2x
y' = -x/9y
now for the tangent line...
y = mx + b
m = -27/9*3
m = -1
3 = -27 + b
30 = b
y = 30 - x
I plugged this answer into one of the answer boxes. It says it's wrong. I tried putting DNE then undefined into both boxes, but it still said the answer was wrong. I have only one try left, so if there's something I'm missing, would somebody please tell me about it?
 A: We want the tangent line(s) to pass through $(27,3)$. As you observed, this point is not on the ellipse $x^2+9y^2=81$.  That's perfectly all right, but it does make the problem harder. The point $(27,3)$ is outside the ellipse, so an informal picture shows there should be two lines through $(27,3)$ that are tangent to the ellipse. 
Let the point of tangency be $(a,b)$. This point is on the ellipse, and therefore 
$$a^2+9b^2=81.\tag{$1$}$$
By implicit differentiation, $2x+18y y'=0$. It follows that the tangent line at $(a,b)$ has slope $-a/9b$.
The equation of the line through $(a,b)$ with slope $-a/9b$ is
$$y-b=-\frac{a}{9b}(x-a).\tag{$2$}$$
The point $(27,3)$ is on this line. It follows that
$$27-b=-\frac{a}{9b}(3-a).\tag{$3$}$$
Simplify Equation $(3)$, and use Equation $(1)$ to find the values of $a$ and $b$. Once you have found these, you will know the equations of the required lines.
A: The problem is correct. There are two distinct tangent lines to the ellipse which pass through the point. The point is not on the ellipse (otherwise, yes, there would be only one). Figuring out the points at which those lines are tangent to the ellipse is part of the problem.
