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/
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