It is known that the moving point P of two points m (4.0) n (1.0) on the plane satisfies the equation PM = 2pn (1) to find the locus C of the moving point P. (2) if the point Q (a, 0) is a point in the locus C, any linear L intersection LOCUS C passes through Q at two points AB, it is proved that the value of vector QA multiplied by vector QB is only related to a; Let f (a) = vector QA multiplied by vector QB to find the value range of F (a)

(1) Let point P coordinate (x, y), then vector PM + (4-x, - y), vector PN + (1-x, - y), the absolute value of vector PM is equal to (4-x) &# 178; + Y & # 178;) under the root sign, from the absolute value of vector PM is equal to 2, the absolute value of vector PN is equal to (4-x) &# 178; + Y & # 178; = 2 under the root sign (1-x & # 178;) + Y & # 178;, the square of x plus the square of Y is equal to four, and its trajectory is a circle with the center at the origin and radius of 2
(2) If K exists, then the linear equation is y = K (x-a), substituting X & # 178; + Y & # 178; = 4,
In this paper, (1 + K & # 178;) x & # 178; - 2ak & # 178; X + (K & # 178; a & # 178; - 4) = 0, let a (x1, Y1), B (X2, Y2), then X1 + x2 = 2ak & # 178; 1 + K & # 178
As a result, QA is not related to K, and it is related to a, so f (a) = a-178; (4) (f (a) = a-178; (4) (f (a) = a-178; (4) because the point Q (a, 0) is a point in the trajectory, SO-2 is less than a, SO-2 is less than a, SO-2 is less than a, SO-2 is less than a, less than a, less than 2,0 is less than a, less than A-2, less than A-2, less than a-2,0 is less less than a-178; less than a-178; (4, - 4 is less less than a-178; (4) is less less than a-178; (4, so f (a) = a-178; - 4, and because point Q (a) is A-4, because point Q (a, q (a) is a point Q (a) is a point in the track, SO-2, SO-2, so-2-2, SO-2, so-2-2 is a point 0, that is, the value range of F (a) = A & # 178; - 4 is (- 4,0)

RELATED INFORMATIONS