tangent lines to the ellipse

General_Sax

1. The problem statement, all variables and given/known data
Find the equations of both the tangent lines to the ellipse x2 + 9y2 = 81 that pass through the point (27, 3).
One is horizontal the other is not.


2. Relevant equations



3. The attempt at a solution
horizontal, easy: y = 3



x^2+9y^2=81
derivative:

2x + 18yy` = 0
y`= -x/9y
at the point (27,3) the slope will be -1.

y-3 = -(x-27)

y= -x + 30

this solution is wrong according to my online assignment program, but I can't for the life of me see why.

Mark44

You are assuming that the point (27, 3) is on the ellipse when you use your formula for y', but this is not a point on the ellipse. I would approach this problem by sketching a graph of the ellipse, and drawing a line from the given point to that it is tangent to the ellipse at some unknown point (x₀, y₀). The slope at the tangent to the ellipse at this point has to be equal to what your derivative says, and also has to be equal to the slope of the line between this point and the point (27, 3).