Find the relation between the angle made by a straight line connecting the origin and an ellipse (angle made between that line and the x axis) and the slope of the tangent to the ellipse. This should also give you the coordinates on the ellipse. In particular at which point/points does the slope of the tangent is $45^{\circ}$? What is the angle the straight line connecting the origin to that point (points) make with the x axis? You can assume the equation of the ellipse is $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,a>b$$.
For reference, for a circle of radius $a$, the slope of the tangent is 45 degrees at the points such that $y=\pm x$. There are 4 such points one in each quadrant. The line connecting the center of the circle to those points also happens to make a 45 degree with the x axis. And the slope of the tangent at those points is also 45 degrees.
My approach would be to find the equation to the tangent of the ellipse, then find the equation of the line connecting the origin to the point $(x,y)$ of the ellipse and do the inner product of the two unit vectors along those two lines to find the angle between them. But then how to relate that to the angle of the line connecting the origin to the points in the ellipse? Or maybe there is an easier way to go about this?