# 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

## 1 Answer

Hint:

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).

Similarly one can treat the case when the tangent point and the center of the circle are in different half-planes.