Given that the equation of circle O is x ^ 2 + y ^ 2-2 = 0, the equation of circle C is x ^ 2 + y ^ 2-8x + 10 = 0, and the tangent length from the moving point P to circle O and circle C is equal, what is the trajectory equation of point P?

Given that the equation of circle O is x ^ 2 + y ^ 2-2 = 0, the equation of circle C is x ^ 2 + y ^ 2-8x + 10 = 0, and the tangent length from the moving point P to circle O and circle C is equal, what is the trajectory equation of point P?

First, we change the two circles into the standard equation: X & sup2; + Y & sup2; = 2, (x-4) & sup2; + Y & sup2; = 6. According to the tangent length and Pythagorean theorem: | op | & sup2; - 2 = | o'p | & sup2; - 6, let P point coordinate (x, y), then x & sup2; + Y & sup2; - 2 = (x-4) & sup2; + Y & sup2; - 6, we get x = 3 / 2, so the point P trajectory equation is a straight line

Four small balls with different colors, such as red and black, are randomly put into three different boxes a, B and C, and each box has at least one small ball
Find the probability of two red and black balls in box a at the same time?
Find the probability of two red and black balls in the same box?

According to the meaning of the question, the colors of the four balls are different, and each box has at least one small ball, which is put in at random. Therefore, two of the four balls are selected as a group, with C (4,2) selection methods, and then randomly put into a, B, C three boxes, with a (3,3) placement methods

Given that 0 ≤ A-B ≤ 2, - 2 ≤ a + B ≤ 0, then the range of a + 3b is______ ．

Let a + 3B = m (a-b) + n (a + b) = (M + n) a + (- M + n) B, ｜ 1 = m + N3 = − m + N, the solution is m = − 1n = 2. ∵ 0 ≤ A-B ≤ 2, - 2 ≤ a + B ≤ 0, ｜ - 2 ≤ - (a-b) ≤ 0, - 4 ≤ 2 (a + b) ≤ 0, ｜ - 6 ≤ a + 3b ≤ 0, the range of ｜ a + 3b is [- 6, 0]. So the answer is: [- 6, 0]