Line Tangent to an Ellipse

Date: 03/29/2003 at 18:21:40
From: Maya
Subject: A line tangent to an ellipse

Find the equation of the tangent to the ellipse x^2 + y^2 = 76 at each 
of the given points. Write your answers in the form y = mx + b.

   a. (8,2) 
   b. (-7,3) 
   c. (1,-5)

My main problem is finding the slope of the line, because since I 
have one point on the line, I need either another point on the line, 
or the slope of the line. I would also like to know how to find the 
slope of the ellipse and how it relates to the slope of its tangent. 
I tried to use the equation of the ellipse [x^2 + y^2 = 76] and 
y = mx + b as a system of equations, but when I substituted the values
for x and y I ended up getting the answers that I already had. And I
don't know how to find the slope of the line.

Date: 03/29/2003 at 20:00:45
From: Doctor Rob
Subject: Re: A line tangent to an ellipse

Thanks for writing to Ask Dr. Math, Maya.

There are two ways to find the slope of the tangent line at a point,
one using calculus, and one not. I'll show you the non-calculus 
approach first.

Let the point on the ellipse be (x0,y0), and the tangent line with
slope m be

   y = m*(x-x0) + y0.

The points in common between this line and the ellipse are the common
solutions of this equation and the equation of the ellipse,

   x^2 + y^2 = 76.

Substitute for y from the first equation into the second,

   x^2 + [m*(x-x0) + y0]^2 = 76.

This is a quadratic equation in x. In general, this has two solutions 
x, which are the x-coordinates of the two places where a line 
intersects an ellipse. We know one of the roots, namely x = x0. That 
means that this quadratic equation,

   (1+m^2)*x^2 + 2*m*(y0-m*x0)*x + (y0-m*x0)^2 - 76 = 0

must have x = x0 as one root, and the other root must also be x = x0.  
Thus the left side must be identically equal to

   (1+m^2)*(x-x0)^2.

That will allow you to solve for m in terms of x0 and y0.

To use calculus, take the equation of the ellipse and differentiate
implicitly with respect to x. Solve the resulting equation for dy/dx, 
substitute x = x0 and y = y0, and you'll have the expression for the 
slope of the tangent line at any point (x0,y0).

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/