# Tangent line to ellipse

1. Mar 5, 2008

### bajazu

Given equation x^2/9 + y^2/4 = 1. Determine the two points on the ellipse having a tangent line passing through the point (0,3). Cant seem to figure this one out? I have found the derivative of the slope which is -4x/9y. I dont know how to use that slope in order to find a tangent line that passes through the points (0,3), which is not located on the orignal ellipse?

Last edited: Mar 5, 2008
2. Mar 5, 2008

### dynamicsolo

Once you have the general expression for the slope of the tangent line to a point (x,y) on the ellipse, you need to look at the slope of a line that runs from (x,y) to (0,3). That will be

m = (y - 3)/(x - 0) ,

which you now equate with the slope for a tangent line (in other words, having found the slope for a line running from any point on the ellipse through (0,3), we now want to find the ones that are tangent lines).

The expression you get from equating these slopes can then be combined with the original equation for the ellipse (by whatever method you find convenient) to solve for either x or y, after which you can then find the remaining coordinate. (If you solve for y first, you'll get a single value, which marks two points on the ellipse.)

Last edited: Mar 5, 2008
3. Mar 5, 2008

4. Mar 5, 2008

### dynamicsolo

What the problem is asking for is to find tangent lines which pass through the external point (0,3) and which are tangent to the given ellipse. The values (x,y) we are solving for are points on the ellipse, so those coordinates must satisfy the equation for the ellipse.

The equation which matches the slopes gives us

(-4x) · x = (9y) · (y-3) , or

$$-4x^{2} = 9y^{2} - 27y$$.

The solutions we are looking for must satisfy both this equation, so that the lines were are dealing with will be tangent to the ellipse, and the equation for the ellipse, since the tangent points are on the ellipse. If you solve this last equation for $$x^{2}$$, you can substitute that into the ellipse equation to solve it for y (you should get two values). Then use either equation to solve for the x-coordinates.

5. Jan 5, 2011