It is known that the parabola y = ax of bungalow + BX + C (a > o) and the straight line y = K (x-1) - K bungalow / 4, no matter K is any real number, the parabola and the straight line have only one common number

It is known that the parabola y = ax of bungalow + BX + C (a > o) and the straight line y = K (x-1) - K bungalow / 4, no matter K is any real number, the parabola and the straight line have only one common number

Y = K (x-1) - K ^ 2 / 4 = KX - (4K + K ^ 2) / 4, substituting y = ax ^ 2 + BX + C (a > o), we get: KX - (4K + K ^ 2) / 4 = ax ^ 2 + BX + CAX ^ 2 + (B-K) x + (4C + 4K + K ^ 2) / 4 = 0, then the discriminant = 0, that is: (B-K) ^ 2 - 4a (4C + 4K + K ^ 2) / 4 = B ^ 2-2bk + K ^ 2-4ac-4ak-ak ^ 2 = 0

If there is only one common point between the straight line y = K (x-1) - K24 and the parabola y = AX2 + BX + C regardless of the value of K, the values of a, B and C are obtained

∵ the straight line y = K (x-1) - K24 and the parabola y = AX2 + BX + C have and only have one common point, ∵ the system of equations: y = K (x − 1) − k24y = AX2 + BX + C has only one set of solutions, ∵ AX2 + (B-K) x + K24 + K + C = 0 has equal real number solutions, ∵ a = 0, ∵ 1-A) k2-2 (2a + b) K + b2-4ac = 0 ∵ for any real number k, the above formula holds, ∵ 1 − a = 0 − 2 (2a + b) = 0b2 − 4ac = 0, ∵ a = 1, B = - 2, C = 1

It is stipulated that x * y = ax + by cxy, A.B.C is a known number, and the right side of the equation is the usual addition and subtraction operation. It is also known that 1 * 2 = 3,2 * 3 = 4, X * m = x (M is not equal to 0) to find M

My idea: first of all, put 1 ^ 2 = 3,2 ^ 3 = 4, x ^ m = x into the first formula to get the following: 1 ^ 2 = a + 2b-2c = 3 -------- ① 2 ^ 3 = 2A + 3b-6c = 4 -------- ② x ^ m = ax + BM CMX = (A-CM) x + BM = x -------- ③ because m is not equal to 0, so from the above formula