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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
Let a (x0, Y0), B (x1, Y1), the linear equation: y = - 2x-2 be substituted into the elliptic equation to solve: 9x * x + 16x + 6 = 0
So x1-x0 = √ (x0 + x1) ^ 2-4x0x1 = 2 √ 10 / 9, if the slope is - 2, then y1-y0 = - 4 √ 10 / 9
s=c（y0-y1）=4√ 10/9
RELATED INFORMATIONS
- 1. 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|
- 2. 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
- 3. 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
- 4. 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______ ．
- 5. 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)
- 6. 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
- 7. 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______ ．
- 8. Let AB be the chord passing through the center of the ellipse and f be a focal point of the ellipse? The ellipse is x ^ 2 + 2Y ^ 2 = 1
- 9. It is known that the eccentricity of ellipse C: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a is greater than B is greater than 0) is root sign 2 / 2, and its left and right focuses are F1F2 respectively. P is a point on the ellipse. Vector Pf1 × vector PF2 = 3 / 4, absolute value of OP = root sign 7 / 2 1. Solving the equation of ellipse C 2. The moving line L passing through point s (0, - 1 / 3) intersects with ellipse C and a, B. question: is there a fixed point m on the y-axis so that a circle with diameter AB passes through point m? If there is, find out the coordinates of M. if not, explain the reason
- 10. The focal length of the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 is 2C. If the three numbers a, B and C form an equal ratio sequence in turn, calculate the eccentricity E
- 11. 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
- 12. 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
- 13. 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!
- 14. 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)
- 15. 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___ ,
- 16. 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
- 17. 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
- 18. 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
- 19. 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
- 20. 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