# Which point on a given line maximizes anglular separation of two given points?

Suppose points $$F_1$$ and $$F_2$$ are the left and right foci of the ellipse $$(x^2/16)+(y^2/4)=1$$, respectively and point $$P$$ is on the line $$L$$ which is $$x-\sqrt3y+8+2\sqrt3=0$$. When $$\angle F_1PF_2$$ reaches the maximum then the value of the ratio $$|PF_1/PF_2|$$

1. $$1 + \sqrt2$$
2. $$\sqrt2 - 1$$
3. $$\sqrt3 - 1$$
4. $$\sqrt3 + 1$$

Ans is C

Solution could be found here

Plz explain the proof of the theorem "euclidean geometry tells us that $$\angle F_1PF_2$$ reaches the maximum only if the circle through points $$f_1 , f_2 , p$$ is tangent to the line $$1$$ at $$p$$."

• As $\angle F_1PF_2$ is inscribed, it's equal to $\frac12$ of the arc $F_1F_2$, not contatining the point $P$. As the chord is fixed, then we need to minimize the radius. Can you show that the minimal radius is obtained when $F_1P$ or $F_2P$ is a diameter? In other words, when $F_1P$ or $F_2P$ is perpendicular to the line and therefore the line is the tangent line at $P$? Jun 10 '20 at 10:09

You are looking for a point $$P$$ lying a line $$\ell$$ which maximizes $$\angle F_1PF_2$$. We will assume that the line does not intersect segment $$F_1F_2$$ (otherwise the largest angle $$\pi$$ is achieved in the point of intersection with the segment).
Draw a circle through the points $$F_1,F_2$$. Observe that there exist two radius values, such that the circle is tangent to the line $$\ell$$. For one of the values the tangent point and the circle center are in the same half-plane created by the line $$(F_1F_2)$$ , and for the other value they are in the different half-planes.
Assume the former case. By the inscribed angle theorem the segment $$F_1F_2$$ will subtend the same angle for all points lying on the circle arc $$F_1F_2$$. The value of the angle is $$\sin\phi=\frac{F_1F_2}{2R}$$, where $$R$$ is the radius of the circle. In the considered case $$0<\phi<\frac\pi2$$. In this part $$\sin\phi$$ is increasing function of $$\phi$$, so that $$\phi$$ is decreasing function of the circle radius. So we need to minimize the radius while circle still intersects the line. By continuity it is achieved if the circle and the line $$\ell$$ intersect in a single (tangent) point (otherwise a part of the line is inside the circle and the radius can be decreased while the circle still intersecting the line).