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Given that two circles C1: (x + 4) 2 + y2 = 2, C2: (x-4) 2 + y2 = 2, the moving circle m is tangent to both circles C1 and C2, then the trajectory equation of the moving circle center m is () A. X = 0b. X22-y214 = 1 (x ≥ 2) C. x22-y214 = 1D. X22-y214 = 1 or x = 0
If two definite circles and moving circles are circumscribed or both inscribed, that is, two circles C1: (x + 4) 2 + y2 = 2, C2: (x-4) 2 + y2 = 2, moving circle m is tangent to both circles C1 and C2, that is, point m is on the vertical bisector of line C1 and C2, and the coordinates of C1 and C2 are (- 4,0) and (4,0) respectively. Its vertical bisector is y-axis, and the trajectory equation of moving circle center m is x = 0 A circumscribe, let it be inscribed with circle C1: (x + 4) 2 + y2 = 2, and circumscribed with circle C2: (x-4) 2 + y2 = 2, then the distance from m to (4,0) is reduced to (- 4,0), and the difference is 22. According to the definition of hyperbola, the locus of point m is a hyperbola with (- 4,0) and (4,0) as the focus and 2 as the length of real half axis, so B2 = c2-a2 = 14, so the equation of hyperbola is x22-y214 = 1 It is known that the trajectory equation of moving circle m is x22-y214 = 1 or x = 0, D
Given that circle C1: x ^ 2 + y ^ 2-2mx + 4Y + m ^ 2-5 = 0, circle C2: x ^ 2 + y ^ 2-2my + m ^ 2-3 = 0, when m is the value, two circles are circumscribed, included, separated and inscribed?
Ha ha, I just did this problem today!
Circle C1: x ^ 2 + y ^ 2-2mx + 4Y + m ^ 2-5 = 0 (x-m) ^ 2 + (y + 2) ^ 2 = 9
Circle C2: x ^ 2 + y ^ 2-2my + m ^ 2-3 = 0 x ^ 2 + (y-m) ^ 2 = 3
Find the center distance D of two circles, and then judge the relationship between D and R + R, r-r
1: |c1c2 | = R + R √ m ^ 2 + (M + 2) ^ 2 = 3 + √ 3 M = - 1 ± √ 5-3 √ 3 circumscribed
2: |c1c2 | = R + R √ m ^ 2 + (M + 2) ^ 2 = 3 - √ 3 M = - 1 ± √ 5 + 3 √ 3 inscribed
3: R-r-1 + √ 5-3 √ 3 or M
Circle C1: x ^ 2 + y ^ 2 + 2mx + 4Y + m ^ 2-5 = 0 circle C2: x ^ 2 + y ^ 2-2x-2my + m ^ 2-3 = 0 (M greater than 0)
Find (1) if the two circles are circumscribed, find out the length of the outer common tangent line of the two circles of m at this time
(2) Is there m to make two circles intersect? If so, find out the range of M. if not, explain the reason
Circle C1: X & # 178; + Y & # 178; + 2mx + 4Y + M & # 178; - 5 = 0, circle C2: X & # 178; + Y & # 178; - 2x-2my + M & # 178; - 3 = 0, (M > 0), that is, the distance between the center of the circle is the sum of the radii of the two circles
Circle C1:
x²+y²+2mx+4y+m²-5=0
x²+2mx+m²+y²+4y+4=9
(x+m)^2+(y+2)^2=3^2
The center of circle C1 is x = - m, y = - 2, r = 3
Circle C2:
x²+y²-2x-2my+m²-3=0
x²-2x+1+y²-2my+m²=4
(x-1)^2+(y-m)^2=2^2
That is, the center of circle C2 is x = 1, y = m, r = 2
We get: (1 + m) ^ 2 + (M + 2) ^ 2 = 5 ^ 2
The solution is m = - 5 or M = 2, because m > 0, so m = 2
The test shows that M = 2 is in accordance with the meaning of the question
The length of the tangent line L = radical ((2 + 3) ^ 2 - (3-2) ^ 2) = 2 radical 6
If there is intersection between circle C1 and circle C2, the center distance should be less than 5 and greater than 1,
That is 1
Why should the center distance of a circle be less than 5 and greater than 1
If the center distance of two circles is equal to the sum of radius of two circles, it should be √ [(1 + m) &# 178; + (M + 2) &# 178;] = 3 + 2 = 5
Square is (1 + m) & 178; + (M + 2) & 178; = 25
It is known that circle C1: x + y-2mx + 4Y + m-5 = 0, circle C2: x + y + 2x-2my + M-3 = 0. When m is the value of (1) circle C1 and C2 circumscribe (2) circle C1 and C2 contain the same problem
Circle 1: (x-m) + (y + 2) = 1, radius 1, circle 2: (x + 1) + (y-m) = 4, radius 2, circumscribed two circles, the distance between the centers of two circles is equal to 3, the equation (M + 1) + (M + 2) = 9, M = the second question, the distance between the centers of two circles is greater than the coincidence of the centers of two circles, and the equation group 0 < 2m + 6m + 5 < 1
RELATED INFORMATIONS
- 1. Given circle C1: (x + 3) ^ 2 + y ^ 2 = 1 and circle C2: (x-3) ^ 2 + y ^ 2 = 9, the moving circle m is circumscribed with circle C1 and C2 at the same time The orbit equation of moving circle center M
- 2. If the parabola C1 and the parabola C2: y ^ 2 = - 4x are symmetric with respect to the straight line x + y = 2, then the focal coordinate of the parabola C1 is
- 3. Let the equation of the ellipse be x square + y square / 4 = 1, the straight line passing through M (0,1) intersects the ellipse at two points AB, O is the coordinate origin, Op vector = 1 / 2 (OA vector + ob vector). When l rotates around point m, the trajectory equation of the moving point P is obtained
- 4. The linear equation with ellipse (x ^ 2 / 9) + (y ^ 2 / 4) = 1 intersecting at two points a and B, and the midpoint of line AB is m (1,1) is, using the point difference method
- 5. If the midpoint of line a and B is (1,1), then the equation of L is?
- 6. It is known that m (4,2) is the midpoint of line AB cut by ellipse x2 + 4y2 = 36, then the equation of line L is______ ．
- 7. A. B is two points on hyperbola x ^ 2-y ^ 2 / 2 = 1, and point (1,2) is a line segment ab. the midpoint of the line AB equation is obtained
- 8. The line passing through point P (8,1) intersects hyperbola x2-4y2 = 4 at two points a and B, and P is the midpoint of line ab. the equation of line AB is obtained
- 9. Given the hyperbola x ^ 2-4y ^ 2 = 4 and the point m (8,1), the straight line passing through the electric m intersects the hyperbola at two points a and B, and M is the midpoint of the line AB, the equation of the straight line is obtained
- 10. If the line and hyperbola x2-4y2 = 4 intersect at two points a and B, if the midpoint coordinate of line AB is (8,1), then the equation of the line is______ ．
- 11. It is known that the line L passes through the point P (- 2,5), and the slope is - 3 / 4. The equation of the circle where the line L is tangent to the point (2,2) and the center of the circle is on the line x + Y-11 = 0 is obtained Who knows? Reply
- 12. If a straight line passes through the point P (- 3, - 32), and the distance between the center of circle x2 + y2 = 25 and the straight line is 3, then the equation of the straight line is___ ．
- 13. The equation of the line parallel to the line 2x-y + 1 = 0 and tangent to the circle x2 + y2 = 5 is () A. 2x-y + 5 = 0b. X2-y-5 = 0C. 2x + y + 5 = 0 or 2x + Y-5 = 0d. 2x-y + 5 = 0 or 2x-y-5 = 0
- 14. Find the equation of a straight line passing through point a (3,4) and tangent to circle x square + y square = 9 Such as the title
- 15. It is known that the ellipse takes the symmetry axis as the coordinate axis, and the major axis is three times of the minor axis, and passes through the point (3,0)
- 16. Find the equation of the circle which passes through the point m (3, - 1) and the circle C: x2 + Y2 + 2x-6y + 5 = 0 and is tangent to the point n (1,2). Why can't we use the equation of the circle system? Isn't there an intersection point between the tangent and the circle?
- 17. A beam of light is emitted from a (- 2,3), reflected by x-axis, and tangent to the circle C (x-3) 2 + (Y-2) 2 = 1. The equation of the straight line of the reflected light is obtained Method 1: the symmetrical point of point a about the x-axis is on the line where the reflected light is located PS: it's a clue to the title, but I'm not good at it
- 18. Find the tangent equation of the circle x 2 + y 2 = 4 from point a (2,4)
- 19. Tangent equation of P (- 5, - 2) and circle x ^ 2 + y ^ 2 = 25 RT
- 20. Find the tangent equation passing through point a (- 4,3) and tangent to circle (x + 1) square + (y + 1) square = 25 Find the tangent equation passing through point a (- 4,3) and tangent to circle (x + 1) square + (y + 1) square = 25