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Let E1 and E2 be the eccentricities of ellipses and hyperbolas with common intersections F1 and F2 respectively, p be a common point, and the segments Pf1 and PF2 are vertical Find the value of (E1 ^ 2 + E2 ^ 2) / (E1E2) ^ 2. (^ 2 is the square)
Very simple, as long as the conditions of the subject are transformed into algebraic expression, and then the result is obtained by evolutionary simplification
Let the long half axis of ellipse be A1, the real half axis of hyperbola be A2, and their half focal length be c
Let Pf1 = m, PF2 = n, M > n. according to the definition of ellipse and hyperbola, we can get
m+n=2a1
m-n=2a2
Solution
m=a1+a2,n=a1-a2
Pf1 ⊥ PF2, obtained from Pythagorean theorem
PF1²+PF2²=F1F2²
(a1+a2)²+(a1-a2)²=(2c)²
How to simplify
a1²+a2²=2c²
Eccentricity E1 = C / A1, E2 = C / A2
(e1²+e2²)/(e1e2)²
=[(c/a1)²+(c/a2)²]/[(c/a1)(c/a2)]²
=[(c²/a1²)+(c²/a2)²]/[c²/(a1a2)]²
=[c²(a1²+a2²)/(a1a2)²]/[c⁴/(a1a2)²]
=c²×2c²/c⁴
=2
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