Given the circle C: x2 + y2-2x + 4y-4 = 0, whether there is a straight line L with slope of 1, so that the chord length ab of L cut by circle C is the diameter of the circle passing through the origin, if there is a straight line equation L, if not, explain the reason

Given the circle C: x2 + y2-2x + 4y-4 = 0, whether there is a straight line L with slope of 1, so that the chord length ab of L cut by circle C is the diameter of the circle passing through the origin, if there is a straight line equation L, if not, explain the reason

The standard equation of circle C is (x-1) 2 + (y + 2) 2 = 9. Suppose there is a circle m with diameter AB, and the coordinates of the center m are (a, b). ∵ cm ⊥ L, that is, KCM · KL = B + 2A − 1 × 1 = - 1 ≁ B = - A-1 ≁ the equation of line L is y-b = x-a, that is, x-y-2a-1 = 0 ≁ cm | 2 = (| 1 + 2 − 2A − 1 | 2) 2 = 2 (1-A) 2 ≁

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, the solution is m ＜ 5; (4 points) (2) let m (x1, Y1), n (X2, Y2), the simultaneous straight line x + 2y-4 = 0 and the equation of circle x2 + y2-2x-4y + M = 0, eliminate y, and get 5x2-8x + 4m-16 = 0

(2x-5y)·_____ =4x^2-25y^2

(2x-5y)（2x+5y）=4x^2-25y^2

If P: 1-xa is not a sufficient and unnecessary condition for Q to be non-p, the value range of a is obtained

Not p is 1-x > = 0, that is, x = 1 is the answer. Sufficient and unnecessary definition: if there is a thing case a, there must be a thing case B; if there is no thing case a but not necessarily no thing case B, a is the sufficient and unnecessary condition of B, referred to as sufficient condition

Given the proposition p: | 1-x-13 | ≤ 2, Q: x2-2x + 1-m2 ≤ 0 (M > 0), and P is a sufficient and unnecessary condition of Q, the value range of real number m is obtained

By the proposition p: - 2 ≤ 1-x-13 ≤ 2, that is - 2 ≤ x-13-1 ≤ 2, ∧ - 1 ≤ X-13 ≤ 3, ∧ - 3 ≤ X-1 ≤ 9, ∧ - 2 ≤ x ≤ 10, by the proposition q: ∫ x2-2x + 1-m2 ≤ 0 (M > 0), ∧ [X - (1-m)] [x - (1 + m)] ≤ 0, ∧ m > 0, ∧ 1-m ≤ m ≤ 1 + m, ∧ P is a sufficient and unnecessary condition for Q, ∧ 1-m ≤

P：『1-(x-1)/3』

p: X > = - 2 is not p XM or X-1

Given that P: | 1 − x − 13 | ≤ 2, Q: x2 − 2x + 1 − M2 ≤ 0 (M > 0), and that non-p is a necessary and sufficient condition for non-Q, then the value range of M is______ ．

From | 1-x − 13 | ≤ 2, we obtain | x-4 | ≤ 6, and the solution is - 2 ≤ x ≤ 10. That is, P: - 2 ≤ x ≤ 10. From x2-2x + 1-m2 ≤ 0, we obtain [x - (1-m)] [x - (1 + m)] ≤ 0, ∵ m > 0, ∩ 1-m < 1 + m, and the solution of the inequality is 1-m ≤ x ≤ 1 + m, that is, Q: 1-m ≤ x ≤ 1 + M

If the vertex coordinates of the parabola y = 2x & # 178; + 8x + C are on the x-axis, then C=

8

When x is 2 and 1.5 respectively, find the value of the algebraic formula 3x (x + 1) / 2

When x = 2
3x（x+1）/2
= 3*2*（2+1）/2
=9
When x = 1.5,
3x（x+1）/2=
3*1.5*（1.5+1）/2
=3*3/2*5/2/2
=45/8.

Given that x ^ 2-3x-2 = 0, find the value of the algebraic formula [(x-1) ^ 3-x ^ 2 + 1] / X-1

Because x ^ 2-3x-2 = 0, x ^ 2-3x = 2
[(X-1)^3-X^2+1]/X-1
=[(X-1)^3-(X^2-1)]/X-1
=(x -1)^2-(x +1)
=x^2-3x
=2