What Are the Tangent Lines to an Ellipse Passing Through a Given Point?

[General_Sax]

Homework Statement

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.

Homework Equations

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]

Homework Statement

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.

Homework Equations

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.

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).