Ellipse and the Right Angle

Geometry Level 4

Below is the graph of the ellipse \(\large\frac{x^2}{16}+\frac{y^2}{9}=1\) and two lines that are tangent to the ellipse and meet at right angles at a point \(P.\) If we trace out the path of \(P,\) its total length can be expressed as \(k\pi.\) What is \(k?\)