It is known that the ellipse and hyperbola have the same focus, and the two curves intersect at P, Pf1 and are perpendicular to PF2. Ask the relationship between the two eccentricities (expressed by E1 and E2 respectively) It is known that the ellipse and hyperbola have the same focus F1 and F2, and the two curves intersect at P, Pf1 is perpendicular to PF2. Ask the relationship between the eccentricity of the two curves (expressed by E1 and E2 respectively)

Using focal radius

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1. A point P on the hyperbola X & # 178; - Y & # 178 / 9 = 1 with F2 as the left and right focus satisfies the vector Pf1 * vector PF2 = 0 |Vector Pf1 | + | vector PF2 | =? Seek the concrete process
2. Let F 1 and F 2 be the left and right focal points of the hyperbola x 2-y 2 / 9 = 1 respectively. If P is on the hyperbola and pf1pf 2 = 0, then | Pf1 + PF2 | =?
3. It is known that P is a point on the right branch of hyperbola x2 / 16-y2 / 9 = 1, and F1 and F2 are the left and right focal points respectively. If | Pf1 |: | PF2 | = 3:2, calculate the coordinates of point P!
4. It is known that the two focal points of hyperbola x ^ 2 / A ^ 2-y ^ 2 = 1 (a > 0) are F1 and F2. P respectively, which are the points on the hyperbola, and ∠ f1pf2 = 90 °, find / Pf1 / * / PF2/
5. It is known that the left and right focus of hyperbola x ^ 2-2y ^ 2 = 2 are F1 and F2 respectively, and the moving point P satisfies the condition | Pf1 | + | PF2 | = 4 Let the locus e of the L intersection of the moving straight line passing through F2 and not perpendicular to the coordinate axis be at two points a and B. ask if there is a point D on the line of2, so that the parallelogram with DA and DB as the adjacent sides is a diamond? Make a judgment and prove it
6. It is known that the sum of the distances between the moving point P and the two focuses F1 and F2 of hyperbola 2x ^ 2-2y ^ 2 = 1 is 4 (1) The equation of trajectory C of moving point P is obtained; (2) If M is the moving point on the curve C, take M as the center and MF2 as the radius to make the circle M. if there are two intersections between the circle m and the Y axis, calculate the value range of abscissa of the point M
7. It is known that the two focal points of hyperbola x ^ 2 / 9-y ^ 2 / 16 = 1 are F1 and F2 respectively, the point P is on the hyperbola, and | Pf1 | x | PF2 | = 32, so the size of angle f1pf2 is calculated
8. F1F2 is the left and right focus of the hyperbola, P is a point on the hyperbola, angle f1pf2 = 60 degrees, triangle pf1f2 = 12 √ 3, and eccentricity is 2
9. It is known that F1F2 is the two focal points of the hyperbola x2 / 4-y2 = 1, and the point P is on the hyperbola and satisfies the angle f1pf2 = 90 ° to find the s triangle f1pf2
10. Let F1F2 be the focus of the hyperbola X / 4 minus y, p be on the hyperbola, and
11. If the ellipse with eccentricity E1 and the hyperbola with eccentricity E2 have the same focus, and the end points of the major axis, the end points of the minor axis and the distance from the focus to an asymptote of the hyperbola form an equal ratio sequence, then (E1 ^ 2-1) / (E2 ^ 2-1)=
12. 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)
13. It is known that F 1 and F 2 are two fixed points, P is the intersection of ellipse and hyperbola with F 1 and F 2 as the common focus, and Pf1 ⊥ PF2, E 1 and E 2 are the eccentricities of the above ellipse and hyperbola respectively, then () A. e12+e22=2B. e12+e22=4C. 1e21+1e22＝2D. 1e21+1e22＝4
14. Known hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 (a > 0b)
15. It is known that the right focus of hyperbola x2a2 − y2b2 = 1 (a > 0, b > 0) is F. if there is only one intersection point between the straight line passing through point F and the right branch of hyperbola with an inclination angle of 60 ° and the right branch of hyperbola, then the value range of eccentricity of hyperbola is () A. （1，2]B. （1，2）C. [2，+∞）D. （2，+∞）
16. It is known that the right focus of hyperbola x2a2 − y2b2 = 1 (a > 0, b > 0) is F. if there is only one intersection point between the straight line passing through point F and the right branch of hyperbola with an inclination angle of 60 ° and the right branch of hyperbola, then the value range of eccentricity of hyperbola is () A. （1，2]B. （1，2）C. [2，+∞）D. （2，+∞）
17. The right focus of hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 is F. if there is only one straight line passing through point F and inclination angle 30 and hyperbola The right focus of hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 is F. if there is only one intersection point between the straight line passing through point F and hyperbola with inclination angle of 30, the range of eccentricity of hyperbola is smaller
18. If the line L passes through the left focus F of hyperbola (x ^ 2) / 3-y ^ 2 = 1 (1) If there is a common point between the line L and the right branch of the hyperbola, point out the range of the inclination angle a of the line L (2) If the line L and hyperbola intersect at two points a and B, P is the midpoint of AB, O is the origin of coordinates, when the slope of OP is equal to 1 / 4, the equation of line L is obtained
19. Given that f is the right focus of hyperbola x216 − Y29 = 1, M is a moving point on the right branch of hyperbola, and a (5,4), the maximum value of 4mf-5ma is______ ．
20. It is known that F1F2 is the left and right focus of hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 (a > 0, b > 0), If the triangle abf2 is an acute triangle, then the value range of eccentricity of the hyperbola