A string 18 units long and its ends are anchored at (2,6) and (8,-2). Find the coordinates of its two vertices?

Question

The length of a string is 18 units and its ends are anchored at (2,6) and (8,-2). An ellipse is traced with a pencil, while keeping the string taut. Sketch a graph of this curve, and find the coordinates of its two vertices.

Answer 1

You are not asked to find the equation of the ellipse (which is very difficult), but only to find the coordinates of its vertices.

Here are some hints.

The center of the ellipse is located at the midpoint of the segment joining the foci. That point is (5,2).

The vertices are 9 units from the center and lie on the line passing through the foci.

1. Find the equation of the line passing through the foci using point-slope--convert to slope intercept.
2. Find the equation of a circle with center at (5,2) with a radius of 9 (since the length of the string is 18 the semi-major axis is 9.
3. Find the points of intersection of the line and the circle. This will give you 2 x-values which are the x-coordinates of the vertices. Find the points of intersection using the substitution method. This is done by substituting the y value (in terms of x) from the line into the y-value for the equation of the circle, resulting in a quadratic equation.

When you have tried all that, here is a graph that can be used to check your work:

desmos.com/calculator/fzhv1xr6n0

Also here is a web site that was used to determine the equation of the ellipse depicted in the Desmos graph:

emathhelp.net/calculators/algebra-2/ellipse-calculator/?type=d&f1x=2&f1y=6&f2x=8&f2y=-2&ma=18