Let a be a moving point on the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1, and the chords AB and AC pass through the focus F1 and F2 respectively. When AC is perpendicular to the X axis, there is exactly | AF1 |: | af2 | = 3:1, (1) find the eccentricity of the ellipse (2) Let AF1 = mf1b, af2 = nf2c, and prove that M + n is the fixed value 6 (all letters are vectors)
1. Let AF1 = 3T, then af2 = t, F1F2 = 2C = 2 √ 2T, that is, C = √ 2T. ① n = 1, this is simple; ② it is proved that M = 5: with the above
RELATED INFORMATIONS
- 1. Let point F1 be the left focus of x ^ 2 / 3 + y ^ 2 / 2 = 1, and the right focus of the chord AB passing through the ellipse. Find the maximum area of triangle f1ab. Remember to find the maximum!
- 2. If the center of the ellipse x ^ 2 / 25 + y ^ 2 / 16 = 1 is a straight line and intersects with the ellipse at two points a and B, and F1 is the focus of the ellipse, then the maximum area of the triangle f1ab is
- 3. Let a straight line passing through the left focus F1 of the ellipse x ^ 2 / 4 + y ^ 2 / 3 = 1 and with an inclination angle of 45 degrees intersect the ellipse and find the perimeter of the triangle abf2 at two points ab
- 4. It is known that the left and right focal points of the ellipse x ^ 2 / 2 + y ^ 2 / 1 = 1 are F1 and F2 respectively. If the straight line passing through point P (0, - 2) and F1 intersects the ellipse at two points a and B, find the triangle ABF It is known that the left and right focal points of the ellipse x ^ 2 / 2 + y ^ 2 / 1 = 1 are F1 and F2 respectively. If the straight line passing through point P (0, - 2) and F1 intersects the ellipse at two points a and B, the area of triangle abf2 is calculated
- 5. Let the center of the ellipse C be at the origin, the focus on the Y axis, and the eccentricity be 2 / 2 of the root sign 1. Find the standard equation of ellipse C 2. If the line L with slope 2 passes through the focus of ellipse C on the positive half axis of Y axis and intersects with the ellipse at two points AB, then | AB is obtained|
- 6. Ellipse x ^ 2 / A ^ 2 + y ^ 2 = 1, triangle ABC takes a (0,1) as the right vertex, B, C on the ellipse, the maximum area of triangle is 27 / 8, find the value of A
- 7. If F1 and F2 are the focus of the ellipse x ^ 2 / 25 + y ^ 2 / 16 = 1, and P is the point on the ellipse that is not on the X axis, then the trajectory equation of the center of gravity g of △ pf1f2 is
- 8. If the hyperbola X29 − y2 = 1 has moving points P, F1 and F2, then the trajectory equation of the center of gravity m of △ pf1f2 is______ .
- 9. It is known that F 1 and F 2 are the left and right focal points of the ellipse C: x 24 + y 23 = 1 respectively, and P is the moving point on the ellipse C, then the trajectory equation of the center of gravity g of △ pf1f 2 is () A. x236+y227=1(y≠0)B. 4x29+y2=1(y≠0)C. 9x24+3y2=1(y≠0)D. x2+4y23=1(y≠0)
- 10. There is a moving point P on the ellipse X / 9 + y2 = 1. F1 and F2 are the two focal points of the ellipse. The trajectory equation of the center of gravity m of △ pf1f2 is obtained
- 11. If AB is the chord passing through the center of the ellipse X & # 178 / 25 + Y & # 178 / 16 and F1 is the left focus, then the maximum area of △ Abf1 is___ 12___ ,
- 12. Let p be a point on the ellipse x ^ 2 / 5 + y ^ 2 / 25 = 1, and F1 and F2 be the two focuses of the ellipse. If Pf1 ⊥ PF2, then the absolute value of the difference between Pf1 and PF2 A.0 B.2√5 C.4√5 D.2√15
- 13. The focus F1, F2 and point P of the ellipse x ^ 2 / 9 + y ^ 2 / 2 = 1 are on the ellipse. If the absolute value of Pf1 = 2 ~ then the absolute value of PF2 = angle f1pf2, the size of PF2 is
- 14. If P is on the ellipse, and (absolute value of Pf1) - (P It is known that the two focal points of ellipse are F1 (0, - 1) F2 (0,1) eccentricity e = 1 / 2 Find 1. Elliptic equation 2. If P is on the ellipse and (absolute value of Pf1) - (absolute value of PF2) = 1, find the cos angle f1pf2
- 15. Given that P is any point on the ellipse x ^ 2 / 4 + y ^ 2 = 1, F1 and F2 are the two focuses of the ellipse, find the minimum value of absolute value Pf1 ^ 2 + absolute value PF2 ^ 2
- 16. If AB is the chord passing through the center of the ellipse x2 / A + Y2 / B2 = 1 and F1 (C, 0) is the focus of the ellipse, then the area of the triangle f1ab is the maximum
- 17. If AB is the chord passing through the center of the ellipse x2 / 25 + Y2 / 16 = 1 and F1 is the focus of the ellipse, then the area of the triangle f1ab is the largest Method 2 (basic method): s △ f1ab = s △ oaf1 + s △ obf1 = (C × y1-y2) × 2 = (3 × y1-y2) × 2 Case (1): when k exists, let AB: y = KX be substituted into the ellipse x2 / 25 + Y2 / 16 to find y1-y2, and get 60 × √ K square × √ [25 × K square × 16] / (25 × K square × 16) < 12 Case (2): when K does not exist, s = B × 2C △ 2 = 12 According to (1) (2), s ≤ 12 My formula is 60 ×√ K square × √ [25 × K square + 16] / (25 × K square + 16), then I get 15,
- 18. Let AB be the chord of the center of the ellipse x ^ 2 / 9 + y ^ 2 / 25 = 1, and f be a focal point of the ellipse, then the maximum area of the triangle ABF is
- 19. Through the focus F of the ellipse x ^ 2 / 4 + y ^ 2 = 1, make the chord AB, and find the maximum area of the triangle AOB (o is the coordinate origin) Ask for detailed explanation
- 20. If the line passing through the origin and the ellipse * * are the left focus of the ellipse, then the maximum area of the triangle ABF is If f (- C, 0) is the left focus of the ellipse, then the maximum area of the triangle ABF is A. The square of BC B, AB C, AC D, B