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Given the curve C: x2 + y2-2x-4y + M = 0. (1) when the value of M is, the curve C represents a circle; (2) if the curve C and the straight line x + 2y-4 = 0 intersect at two points m and N, and OM ⊥ on (o is the origin of the coordinate), find the value of M
(1) From D2 + e2-4f = 4 + 16-4m = 20-4m ＞ 0, m ＜ 5; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; (4 points) (2) let m (x1, Y11, n (X2, Y2) set M (x1, Y11, Y1, n (X2, Y2), the straight line x + 2y-4 = 0, the equation of the circle, the equation x2 + y2-2-2x-4y + M = 0, and the elimination of Y, we get: 5x2-8x2-8x + 4x + 4m-16 = 0, by Weida's theorem, we get: X1 + x2 = 85 (1) by Weida's theorem, the following theorem: X1 + x2 = 85 (1, x1 \: (1): (1): (1) from Weida's theorem, we get: X1 + x2 = 85 (1) 1, X1 : (1 \: (1 \\\\\\\\o x2 = 1: 1: 1: 1: 1= 0, substituting ① and ② into In the above formula, M = 85, and the test result is △ 0, so m = 85 is required
The x-axis of the line L intersecting with the curve C: X ＾ 2 + y ＾ 2-2x-2y + 1 = 0 is known. The y-axis is at two points a and B. o is the origin. The absolute value OA = a, the absolute value ob = B (a is greater than 2, B is greater than 2)
(1) Verification: (A-2) (b-2) = 2
(2) Finding the trajectory equation of the midpoint of line ab
(3) Minimum AOB area of triangle
(1) From the curve C: x ^ 2 + y ^ 2-2x-2y + 1 = 0, the curve C equation can be rewritten as (x-1) ^ 2 + (Y-1) ^ 2 = 1 ^ 2, so C is a circle with (1,1) as the center and 1 as the radius
The equation of the line AB can be written as Y / B + X / a = 1 and reduced to BX + ay AB = 0
The distance d from the center of the circle to the straight line AB = (a * 1 + b * 1-ab) absolute value / root sign (a ^ 2 + B ^ 2) = 1
It is reduced to (A-2) (b-2) = 2
(2) Let the midpoint coordinate of line AB be (x, y). From the midpoint coordinate formula, we can get: x = A / 2, y = B / 2
So substituting a = 2x B = 2Y into (A-2) (b-2) = 2, the AB midpoint equation is (x-1) (Y-1) = 1 / 2
(3) From 2 = (A-2) (b-2) = ab-2 (a + b) + 4, ab-2 (a + b) + 2 = 0 ≥ ab-2 * radical AB + 2 = 0
It is reduced to ab ≤ 6 + 4 * radical 2
So AOB = 1 / 2Ab ≤ 1 / 2 (6 + 4 * radical 2) = 3 + 2 * radical 2
The minimum area of triangle AOB is 3 + 2 * radical 2
If the line 2ax-by + 2 = 0 bisects the circumference of the circle X & sup2; + Y & sup2; + 2x-4y + 1 = 0, then the maximum value of a · B is
Bisecting the circumference of a circle is passing through its center
(x+1)²+(y-2)²=4
Substituting (- 1,2) into 2aX by + 2 = 0
We get a + B = 1
∴ab≤(a+b)²/4=1/4
If circle C1: x ^ 2 + y ^ 2-2ax + 4Y + A ^ - 5 = 0 intersects circle C2: x ^ 2 + y ^ 2 + 2x-2ay + A ^ 2-3 = 0, (1) find the linear equation of intersecting chord; (2) find the chord length of intersecting two circles when a = 0
1) Subtraction:
(1+a)x-(a+2)y+1=0
Linear system x-2y + 1 + a (X-Y) = 0
Constant crossing point (1,1) of x-2y + 1 = 0 and X-Y = 0
The linear equation of intersecting chord is the linear system of constant crossing point (1,1): (1 + a) x - (a + 2) y + 1 = 0
2）a=0
Intersecting chord x-2y + 1 = 0
X ^ 2 + y ^ 2 + 4y-5 = 0 and circle C2: x ^ 2 + y ^ 2 + 2x-3 = 0
c2(-1,0),r2=2
C2 to intersecting chord x-2y + 1 = 0 distance D
d=|-1+1|/√（1^2+2^2)=0
The intersecting chord is C2 diameter
Intersection chord length = 2r2 = 4
RELATED INFORMATIONS
- 1. Given that the circle m is tangent to the x-axis, the center of the circle is on the straight line x-2y = 0, and the maximum distance from the moving point on the circle m to the y-axis is 3, find the square of the circle M
- 2. Circle x ^ + y ^ = 1 and circle x ^ + y ^ - 2x-2y = 0, the position relation is? (^ is 2, that is, x square or Y Square) So R + r = 1 + radical 2, but I calculate that the distance between the two center points is = radical 2. Why is it tangent??? The distance between the center of the circle is less than the sum of the radii, so it intersects!!!!!
- 3. It is known that the straight line X-Y + 3 = 0 and the circle x + y = 1. Judge their position relation and prove it
- 4. Given that P is any point of the circle C: x ^ 2 + y ^ 2 + 4x + ay-5 = 0, the symmetric point of P about 2x + Y-1 = 0d is still on the circle
- 5. Ask for help high school "basic inequality" problem: known circle C: the square of X + the square of Y + BX + ay-3 = 0 (a, B are positive real numbers) on any point about the straight line L: 1 Given that any point on the circle C: x square + y square + BX + ay-3 = 0 (a, B are positive real numbers) with respect to the symmetric point of the line L: x + y + 2 = 0 is on the circle C, then the minimum value of 1 / A + 3 / B is___ .
- 6. Let the center of the circle x ^ 2 + y ^ 2-4x + 2y-11 = 0 be a and the point p be on the circle, then the trajectory equation of the midpoint m of PA is
- 7. (2010` Guangzhou the first mock exam) the known moving circle C passes through point A (-2,0) and is cut in with the circle M: (X-2) ^2+y^2=64. (1) Find the trajectory equation of the center of moving circle C (2) Let L: y = KX + m (where k, m ∈ z) and (1) intersect at two different points B and D, and hyperbola (x ^ 2) / 4 - (y ^ 2) / 12 = 1 intersect at two different points e F. ask if there is a straight line L such that vector DF (vector) + be (vector) = 0 (vector). If there are, please point out how many such lines. If not, please explain the reason
- 8. Given the circle C: X & # 178; + Y & # 178; = 4, the tangent equation of the circle passing through the point (- 2,1) is? What is the answer and the difference between the tangent equation and the ordinary equation?
- 9. The equation of the circle whose center is C (3, - 5) and tangent to the line x-7y + 2 = 0 is______ ．
- 10. Given that the moving circle is circumtangent to the circle (x + 2) 2 + y2 = 4 and tangent to the straight line x = 2, the trajectory equation of the center P of the moving circle is obtained
- 11. It is known that P (x, y) is a moving point on the circle C: x + y-2y = 0. If x + y + m ≥ 0 is constant, the value range of real number m is obtained
- 12. In the equation 3x ay = 8, if x = 3Y = 1 is one of its solutions, then the value of a is______ ．
- 13. If the projection of origin o on line L is point H (- 2,1), then the equation of line L is Please don't use the slope method. We haven't learned that yet. Use the normal vector or direction vector method
- 14. When ma * MB is the minimum value, the equation of line L is obtained
- 15. If the area of △ AOB (o is the coordinate origin) is 2, then the value of B is?
- 16. Given that the point P (m, n) is on the straight line x = - 1, and the point P is symmetrical about the origin of the coordinate is on the straight line y = 3, the coordinate of P is obtained
- 17. When m is equal to what, the inequality mx5
- 18. Known inequality (MX)_ 1) The solution set of (x + 2) > 0 is - 3
- 19. Given that the line x + 2y-3 = 0, the intersection circle x ^ 2 + y ^ 2 + x-6y + F = 0 is at the point P, Q, O, and OP is perpendicular to OQ, then f is?
- 20. If the distance between the point P (m, 3) and the line 4x-3y + 1 = 0 is 4, and the point P is in the inequality 2x + y