The task is to construct the tangent line to the ellipse shown in the figure

The ellipse was the first curve for which Descartes constructed the tangent. His idea was to find a circle that passes through E and center (0,p) (he used the letter v) on the axis. Such a circle will intersect the curve at another point in the neighbourgood of E, but if the radius is normal to the the curve (ellipse in this case) the circle and the curve will have a double point of intersection (xE,yE). Having found the circle the radius will be the normal he was looking for and then the tangent line would be perpendicular to the normal at E. We can then find the slope of the tangent and with it its equation