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Let a straight line passing through the focus of the parabola y2 = 2px (P > 0) intersect with the parabola. Let the coordinates of the two intersections be a (x1, Y1) and B (X2, Y2) respectively. Prove: (1) y1y2 = - P2 (2) x1x2 = p24
It is proved that: (1) let the linear equation be x = my + P2, and substitute y2 = 2px, y2-2mpy + P2 = 0, y1y2 = - P2 (2) x1 · x2 = y12p · y22p = p24
The focus of the parabola y2 = 2px is f, and the coordinates of points a, B and C are (1,2) on this parabola. If point F is exactly the center of gravity of △ ABC, then the equation of line BC is ()
A. 2x+y-1=0B. 2x-y-1=0C. x-y=0D. x+y=0
∵ parabola y2 = 2px, point a (1,2) is in this parabola, ∵ parabola equation y2 = 4x, and f (1,0) can be set B (B2, 2b), C (C2, 2C). From the "two-point equation", the equation of straight line BC is (B + C) y-2bc = 2x. From the problem, point F is just the center of gravity of △ ABC, we can get: 3 = 1 + B2 + C2, 0 = 2 + 2
One focus of the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is f (1,0), and O is the coordinate origin. Let the line L passing through point F intersect the ellipse at two points a and B. if the line L rotates around point F arbitrarily, there will always be OA ^ 2 + ob ^ 2
RELATED INFORMATIONS
- 1. It is known that the center of ellipse C is at the origin and the focus is on the x-axis. The coordinates of a vertex B of ellipse C are (0,1) and the eccentricity is equal to 22. The line L with slope 1 intersects ellipse C at two points m and n. (1) find the equation of ellipse C; (2) ask whether the right focus F of ellipse C can be the center of gravity of △ BMN? If possible, find out the equation of line L; if not, explain the reason
- 2. If P (2,2) is the midpoint of AB, then the equation of parabola C is______ ．
- 3. The left focus of the ellipse is (√ 3,0), and the right vertex is d (2,0). Let a (1,1 / 2) be a moving point on the ellipse. If P is a moving point on the ellipse, find the trajectory equation of the midpoint m of the line PA
- 4. Given that the distance from a point P on the parabola y2 = 16x to the x-axis is 12, then the distance from P to the focus f is equal to 12______ ．
- 5. As shown in the figure, the straight line AB; CD intersects at the point O, OE, bisecting the angle AOC, and the angle BOC - ∠ BOD = 18 ° to find the degree of ∠ BOE
- 6. As shown in the figure, points a, O and B are on the same straight line, ∠ AOC = 1 / 2 ∠ BOC + 30 °, OE bisects ∠ BOC, and calculates the degree of BOE I don't know how to draw or dictate a straight line. The right side is B and the left side is a. there is o point in the middle, with o as the point, extending from the upper right corner to e and from the upper left corner to C~
- 7. As shown in the figure, it is known that OE is the bisector of ∠ AOB, and C is a point in ∠ AOE. If ∠ BOC = 2 ∠ AOC, ∠ AOB = 114 °, calculate the degree of ∠ BOC. ∠ EOC
- 8. As shown in the figure, the straight lines AB and CD intersect at the points o, OE and of, which are bisectors of ∠ AOC and ∠ AOD respectively. It is known that ∠ BOC = 70 ° to find the degree of ∠ DOF and ∠ Coe
- 9. As shown in the figure, it is known that OD bisects BOC and OE bisects AOC, if BOC = 70 ° and AOC = 50 ° (1) Calculate the degree of AOB and its complement angle (2) Request the degree of ∠ doc and ∠ AOE, judge whether ∠ DOE and ∠ AOB are complementary, and explain the reason The DOA is a right angle, 90 degrees, and the diagram is drawn by myself
- 10. Given the angle AOB = 130 degrees, OE bisecting angle BOC, of bisecting angle AOC, calculate the degree of angle EOF
- 11. How to find the directrix and focus of parabola? How to find the known parabola, collimator and focus? It is how to deduce the focal point and collimator of a parabola, or how to deduce a parabola from the focal point and collimator, just as the elliptic standard equation is derived by arithmetic and algebra.
- 12. Let a (x1, Y1), B (X2, Y2), then x1x2 =? Let a (x1, Y1) and B (X2, Y2) pass through the focus of the parabola (x ^ 2) = 4Y to make the chord AB, then what is X1 times x2? There is a process or a graph.
- 13. If the distance between a point P and its focus on the parabola x ^ 2 = 4Y is 2, then the distance between point P and its directrix is 2
- 14. If the distance from a point m on the parabola x ^ 2 = - 16y to the focus is 6, then the coordinate of M is
- 15. Given that the distance from the point P on the parabola x2 = 4Y to the focus is 10, then the coordinate of point P is 0______ ．
- 16. If a ＜ 0, the vertex of the parabola y = x * + 2aX + 1 + 2A * is in the? Th quadrant
- 17. When a
- 18. It is known that the vertex m of the parabola y = AX2 + BX + C with downward opening is in the second quadrant And through the points a (1,0), B (0,1) (1) Please judge the value range of real number a and explain the reason (2) Let the other intersection of the parabola and the x-axis be c. when the area of △ AMC is 25 / 16 times of the area of △ ABC, a is obtained
- 19. Generally, we can use formula to find the vertex and symmetry axis of parabola y = ax ^ 2 + BX + C (a ≠ 0). Y = ax ^ 2 + BX + C = a [x + (B / 2a)] ^ 2+ Generally, we can find the vertex and symmetry axis of parabola y = ax ^ 2 + BX + C (a ≠ 0) y=ax^2 + bx + c =a[x+(b/2a)]^2 + (4ac-b^2)/4a, y=ax^2 + bx + c It should not be converted to (x + B / 2a) ^ 2 = B ^ 2-4ac / 4A ^ 2. How can it be converted to the above form?
- 20. The vertex of the parabola y = AX2 + BX + C is on the Y-axis and passes through two points - (- 1,3), (- 2,6)