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

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

X + y + m > = 0 constant can be changed into m > = - (x + y) constant only needs m to be greater than or equal to the maximum value of - (x + y), so this problem is transformed into the problem of finding the value range of X + y. method 1: let x + y = a, that is, x = a-y be substituted into the circular equation, We get 2Y - (2a + 2) y + a = 0. From the discriminant ≥ 0, we can get 4A + 8A + 4-8a ≥ 0, 1 - √ 2 ≤ a ≤ 1 ＋ 2, that is - 1 - √ 2 ≤ - (x + y) ≤ - 1 ＋ 2  M ≥ - 1 ＋ 2 method 2: let x = cost, Y-1 = Sint, t ∈ R - (x + y) = - cost-sint-1 = - √ 2Sin (T + π / 4) - 1 ≤ √ 2-1  m ＞ 2-1

Given the curve L: the square of X + the square of Y - 2x-4y + M = 0
1) What is the value of m when l is a circle?
2) When m = 4, find the length of line segment of L-section of curve X + 2y-4 = 0

1、
(x-1)²+(y-2)²=-m+1+4
Circle then R & # 178; = - M + 1 + 4 > 0
m

Given the curvilinear talent: x square + y square - 2x-4y + M = 0 intersects with the straight line y = 4-2x, intersects with m and N, and calculates the value of M

X square + y square - 2x-4y + M = 0 (x-1) square + (Y-2) square + m-5 = 0. Take y = 4-2x into the above formula and simplify x square - 2x + m / 5 = 0. Because it intersects two points, it has two different roots 2 square - 4m / 5 > 0 M