find the vertices of an equilaterall triangle circumscribed about the ellipse 9x^2 + 16y^2 = 144.

There can be two such triangles, One with apex up and one with apex down.

The ellipse has semi-major axis length of 4 and semi-minor axis length of 3.



Upper half of ellipse has equation:





We know this slope must be

because the equilateral triangle has angles of 60 degrees, and

. But the line on the right has negative slope.



.

Thus, the line is tangent to the ellipse at

Using we can solve for the lower right vertex of the triangle.

, where a is the x coordinate of this vertex.

Sub in the newly found x coordinate of the tangent point, , and we get:



This means the left vertex has coordinates:



The apex has coordinate:







There is also a triangle that can be positioned with apex down as well.