Minimizing the area of a triangle

Question

Look at the curve x^2 + 3y^2 = 1. In the first quadrant draw different tangent lines to points of the curve. These tangent lines form triangles with x- and y- axes. For which point on the curve is the area of this triangle minimized?

Answer 1

Hi Zac,

This is a challenging problem. There might be an easier technique, but I just took the first strategy that came to mind. I used (a, f(a)) as the point on the curve for which to generate the equation of the tangent line (where f(x) is the function defined by the ellipse in the first quadrant). Once you have the equation of the tangent line, find the x and y intercepts. Then you can create a function that gives the area of the triangle. Finally find the derivative of that function, set equal to zero, and locate the critical points.

This graph has many of the details (but not all of them).

https://www.desmos.com/calculator/zhlpxyf572

Hopefully this answer will be published by WyzAnt--there have been some issues (at least for me) when including links to Desmos URLs in responses.