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How to prove that the sum of distances between any point and two focal points of an elliptic equation is 2a by using the distance formula between two points? The elliptic equation is (x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1) No soy sauce!
The process of deducing elliptic equation in textbook can be reversed
Let F1 (- C, 0), F2 (C, 0), B & sup2; + C & sup2; = A & sup2;, P (x, y) be a point on the ellipse;,
To prove that | Pf1 | + | PF2 | = 2A
Only | Pf1 | & sup2; = (2A - | PF2 |) & sup2;
Only | Pf1 | & sup2; = 4A & sup2; - 4A | PF2 | + | PF2 | & sup2;
Only 4A | PF2 | = 4A & sup2; - | Pf1 | & sup2; + | PF2 | & sup2; (where - | Pf1 | & sup2; + | PF2 | & sup2; = - 4cx)
Only 4A | PF2 | = 4A & sup2; - 4cx
Only a | PF2 | = A & sup2; - Cx
Only | PF2 | & sup2; = (a - CX / a) & sup2;, and (a - CX / a) & sup2; = A & sup2; - 2cx + C & sup2; X & sup2; / A & sup2;
Only (x-C) & sup2; + Y & sup2; = A & sup2; - 2cx + C & sup2; X & sup2; / A & sup2;
Just X & sup2; + C & sup2; + Y & sup2; = A & sup2; + C & sup2; X & sup2; / A & sup2;,
Only X & sup2; (1-C & sup2. / A & sup2;) + Y & sup2; = A & sup2; - C & sup2;, and B & sup2; + C & sup2; = A & sup2;
Only X & sup2; B & sup2; / A & sup2; + Y & sup2; = B & sup2; (both sides are divided by the right term to form an elliptic equation)
From the elliptic equation, we know that the final form is true, so the conclusion is true
Is the sum of the distances from the intersection of ellipse and coordinate axis to the two focal points equal to 2A?
This question is incomplete. You can understand it
(1) When the intersection point is the endpoint of the minor axis of the ellipse, the conclusion is 2A;
(2) When the intersection point is the end point of the major axis of the ellipse, the conclusion is 2C
There is a focal point on the x-axis, which is perpendicular to the two ends of the minor axis. The focal length is 6. The equation for solving the ellipse is given
A focal point on the x-axis is perpendicular to the two ends of the minor axis, and the distance between the focal point and the corresponding directrix is 2
According to the meaning of the title
B = C in this question
You can tell by drawing
and
a²/c-c=2
a²=b²+c²=2c²
that
2c²/c-c=2
2c-c=2
c=2
b=2
a²=b²+c²=8
Equation: X & # 178 / 8 + Y & # 178 / 4 = 1
RELATED INFORMATIONS
- 1. It is known that the length of the minor axis of the ellipse e is 6, and the distance from the focus f to one end of the major axis is 9, then the eccentricity of the ellipse e is equal to () A. 35B. 45C. 513D. 1213
- 2. If the sum of the distances from a point on the ellipse to two focal points (- 4,0), (4,0) is equal to 10, then the length of the minor axis of the ellipse is
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- 4. The focus of the ellipse x ^ 2 / 12 + y ^ 2 / 3 = 1 is F1, F2, and the point P is on the ellipse. If the midpoint of Pf1 is on the Y axis, then the ratio of Pf1 to PF2 is? Mathematical
- 5. The two focal points of ellipse C: X * 2 / A * 2 + y * 2 / b * 2 = 1 (a > b > 0) are F1 and F2. The point P is on ellipse C, and Pf1 ⊥ F1F2, Pf1 = 4 / 3, PF2 = 14 / 3 (1) Solving elliptic c-equation (2) If the line L passes through the center m of the circle x * 2 + y * 2 + 4x-2y = 0, the intersection ellipse C lies at two points a and B, and a and B are symmetric with respect to point m, the equation of the line L is obtained
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- 7. Is the focal length of hyperbola 2C or C
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- 10. 1/2xy+y2+1+x2-1/2xy-2xy-1
- 11. Why is the shortest distance from the focus of the ellipse to the end of the minor axis?
- 12. How to find the shortest distance between focus and ellipse? How to judge which point on the ellipse has the shortest distance to the focus? How to find the shortest distance from the point to the focus?
- 13. When the angle f1pf2 is the largest, the tangent of the angle pf1f2 is 2, Find the size of eccentricity
- 14. If P is a point on the ellipse, the vector Pf1 × vector PF2 =? Needs a general conclusion, and F1F2 is the focus
- 15. If the left and right focus of the ellipse X & # 178 / 4 + Y & # 178 / 3 = 1 is F1F2 and P (x, y) is on the ellipse, then the value range of vector Pf1 * vector PF2?
- 16. Given that P is any point of hyperbola x ^ 2 / 2-y ^ 2 = 1, find the minimum distance between point a (m, 0) (M > 0) and point P?
- 17. Given that the ellipse x2a2 + Y29 = 1 (a > 0) and hyperbola x24 − Y23 = 1 have the same focus, then the value of a is () A. 2B. 10C. 4D. 10
- 18. P is any point on the hyperbola X / 16-y / 9 = 1 which is different from the vertex. F1F2 is the two focuses of the hyperbola. The trajectory equation of the center of gravity of △ pf1f2 is obtained
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