When the angle f1pf2 is the largest, the tangent of the angle pf1f2 is 2, Find the size of eccentricity

The cosine theorem is generally considered for this problem. Cos ∠ f1pf2 = (f1p ^ 2 + F2P ^ 2-f1f2 ^ 2) / (2f1p * F2P) = [(f1p + F2P) ^ 2-2f1p * f2p-f1f2 ^ 2]) / (2f1p * F2P) = - 1 + 4 (a ^ 2-C ^ 2) / (2f1p * F2P). 1 & because f (x) = cosx is a decreasing function when x ∈ (0, π), so ∠ f (x) = cosx is a decreasing function when x ∈ (0, π)

Given that the ellipse X & # 178; / 36 + Y & # 178; / 9 = 1, F1F2 is the left and right focus, if a point P on the ellipse satisfies ∠ f1pf2 = 30 degree,
Finding s triangle f1pf2

Ellipse X & # 178; / 36 + Y & # 178; / 9 = 1A & # 178; = 36, B & # 178; = 9 # C & # 178; = 27 method (1): directly apply the formula s = B & # 178; Tan (30 ° / 2) = 9 * Tan 15 ° = 9 (2 - √ 3) method (2) if you have not learned this formula, let Pf1 = m, PF2 = n, use ellipse to define m + n = 2A = 12

The relationship between the distance from a point on an ellipse to the guide line and the distance to the focus

Ellipse is a kind of conic curve. Now there are two definitions in high school textbooks
1. The sum of the distances from two points on the plane is the set of fixed points (the fixed value is greater than the distance between two points) (these two fixed points are also called the focus of the ellipse, and the distance between the focus is called the focal length);
2. The set of points on the plane whose ratio of the distance to the fixed point and the distance to the fixed line is constant (the fixed point is not on the fixed line, the constant is a positive number less than 1) (the fixed point is the focus of the ellipse, and the line is called the directrix of the ellipse). These two definitions are equivalent
According to definition 2, the distance from a point on the ellipse to the focus is e (eccentricity = C / a) times of the distance to the corresponding guide line

Formula of distance from any point on ellipse to focus
If the eccentricity is known as e, the distance between any point on the ellipse and two focal points on the ellipse can be obtained

ex+a.

When the angle f1pf2 is the largest, the tangent of the angle pf1f2 is 2, Find the size of eccentricity