![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 0 72 71" width="72"></svg>)
Given that the point m is on the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1, the circle centered on M is tangent to the X axis and the right focus F of the ellipse, if the circle m is compared with the Y axis A. B two points, and the triangle ABM is an equilateral triangle with side length 2, the equation for solving ellipse
It can be concluded that m (√ 3,2) f (√ 3,0) C ^ 2 = 3 B ^ 2 = a ^ 2-C ^ 2
Then the coordinates of m point are substituted into x ^ 2 / A ^ 2 + y ^ 2 / (a ^ 2-C ^ 2) = 1
And because a ^ 2 > b ^ 2
So a ^ 2 = 9, B ^ 2 = 5
That is x ^ 2 / 9 + y ^ 2 / 5 = 1
The steps to solve the problem are roughly like this. Let's think about it again. The first time I answered on the Internet, it was a bit crude. I'm sorry
It is known that an ellipse with the center at the far point and the focus on the x-axis intersects the circle X & sup2; + Y & sup2; - 4x-2y + 2.5 = 0 with two points a and B. AB is exactly the diameter of the circle
The slope of AB is - 1 / 2
The equation of circle: X & sup2; + Y & sup2; - 4x-2y + 2.5 = 0
(x－2)²＋(y－1)²＝2.5
The center of the circle is (2,1), and the radius is √ 10 / 2
The slope of line AB is - 1 / 2
The linear AB: Y-1 = - 1 / 2 (X-2)
y＝－1/2x＋2
Let a (XA, - 1 / 2xa + 2), B (XB, - 1 / 2xb + 2), then
The midpoint of AB is the center of the circle (2,1)
xa＋xb＝4
And | ab | = 2R = √ 10, i.e
|AB|²＝(xa－xb)²＋(－1/2xa＋2＋1/2xb－2)²＝10
|xa－xb|＝2√2
Since a and B are equivalent and can be transposed, let XA > XB, then
xa＋xb＝4
xa－xb＝2√2
The solution is: XA = 2 ＋ 2, XB = 2 - √ 2
Then a (2 ＋ 2,1 － 2 / 2), B (2 － 2,1 ＋ 2 / 2)
Let X & sup2; / A & sup2; + Y & sup2; / B & sup2; = 1, M = 1 / A & sup2;, n = 1 / B & sup2;, then the equation is reduced to MX & sup2; + NY & sup2; = 1
Substituting a and B coordinates into the equation, we can get the following results
(2＋√2)²m＋(1－√2/2)²n＝1
(2－√2)²m＋(1＋√2/2)²n＝1
The solution is: M = 1 / 12, n = 1 / 3
A: the elliptic equation is X & sup2 / 12 + Y & sup2 / 3 = 1
If the line y = x + B passes through the center of the circle x ^ 2 + y ^ 2-4x + 2y-4 = 0, then B = how much
Circle x ^ 2 + y ^ 2-4x + 2y-4 = 0
It can be changed into:
(x-2)^2+(y+1)^2=9
So the center of the circle is: (2, - 1)
In this way:
If the point is on a straight line, there are:
-1=2+b
So:
b=-3
RELATED INFORMATIONS
- 1. Find the linear equation of the longest and shortest chord of a point P (3,1) in the circle X & sup2; + Y & sup2; - 8x-2y + 8 = 0
- 2. The straight line passing through the point P (- 2,3) is cut by the circle X & sup2; + Y & sup2; - 4x + 2y-2 = 0, and the linear equation of the longest chord is obtained
- 3. Given the circle x2 + Y2 + KX + 2Y = - K2, when the area of the circle is the largest, the coordinate of the center of the circle is______ ．
- 4. Given that the line L: x-2y-5 = 0 and the circle C: x + y = 50, find (1) the coordinates of intersection a and B, and (2) the area of △ AOB Such as the title, to complete the thinking and process
- 5. If the equation of a circle is the square of X, the square of Y, KX 2Y, and the square of k = 0, then the coordinates of the center of a circle are formula x^2+y^2+kx+2y+k^2＝0 (x+k/2)^2+(y+1)^2=1-(3/4)k^2 The circle has the largest area That is, the maximum radius That is, 1 - (3 / 4) k ^ 2 is the largest Then k = 0 Center (0, - 1) radius 1 X ^ 2 + (y + 1) ^ 2 = 1 1 - (3 / 4) k ^ 2 max Then how can we get k = 0
- 6. If the equation x square + y square + 2mx-2y + m square + 5m = 0 represents a circle, find the value range of real number m? Who knows?
- 7. Given that circle C1: x2 + y2-2mx + M2 = 4, circle C2: x2 + Y2 + 2x-2my = 8-m2 (M > 3), the positional relationship between the two circles is () A. Intersection B. inscribed C. circumscribed D. separation
- 8. If the point P (a, b) is a common point of the straight line x + y = 1 / M and the circle x2 + y2 = 2 / M + 1 / m2, then the value range of AB is
- 9. Given that m (x, y) is any point on the circle x ^ 2 + y ^ 2 = 1, find the value range of Y / (x + 2) This problem appears in the section of standard equation of circle. Thank you here
- 10. Given p (1,2) and circle C: x2 + y2 = 9, make two mutually perpendicular chords through p to intersect C at a and B, and find the trajectory equation of the midpoint of line ab
- 11. Given that the line L: x + Y-9 = 0 and the circle M: 2x & sup2; + 2Y & sup2; - 8x-8y-1 = 0, the point a (4, Y0) is on the line L, B and C are two points on the circle M In the triangle ABC, ∠ BCA = 45 ° AB passes through the center of circle m, then the distance from the center of circle m to the straight line AC is
- 12. Given the circle m; 2x * x + 2Y * y-8x-8y-1 = 0 and the straight line L: x + Y-9 = 0, make a triangle ABC through a point a on the straight line L to make the angle BAC = 45 degrees, AB passes through the center m, and B, C are on the circle M 1: When the abscissa of a is 4, the equation of line AC is obtained 2: Find the value range of abscissa of point a
- 13. If all the points on the curve C: x2 + Y2 + 2ax-4ay + 5a2-4 = 0 are in the second quadrant, then the value range of a is______ ．
- 14. If two tangents of circle (x-1) × (x-1) + y × y = 1 can be made through points (5a + 1,12a), then the value range of real number a is
- 15. Given that the point P (1,2) is on the circle C: X & # 178; + Y & # 178; - 2kx-2y + K & # 178; = 0, then the positional relationship between O (0,0) and circle C is
- 16. Given the circle C: (x-1) 2 + (y + 1) 2 = 1, then the distance from the center of the circle C to the origin of the coordinate o is___ ．
- 17. It is known that the equation of circle C is (x-a) 2 + (Y-A + 1) 2 = 1. When the distance between the origin and the center of circle C is the smallest, the value of a is obtained
- 18. Given that the equation of a circle is X & sup2; + Y & sup2; - 4x + 4Y + 4 = 0, then the shortest distance from the point on the circle to the origin is and the longest distance is
- 19. What is the distance between the center of the circle x ^ 2 + y ^ 2-4x + 2Y + 1 = 0 and the straight line X-Y + 1 = 0
- 20. If there are only three points on the circle (x-1) ^ 2 + (y + 1) ^ 2 = 4 and the distance from them to the straight line 4x + 3Y = m is 1, then the value of M is?