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1. A straight line is cut by two straight lines L1: 4x + y + 6 = 0, L2: 3x-5y-6 = 0, and the midpoint of the line segment is P point. When the coordinate of P point is (0,0), find the straight line 1. A straight line is cut by two straight lines L1: 4x + y + 6 = 0, L2: 3x-5y-6 = 0, and the midpoint of the line segment is P point. When the coordinate of P point is (0,0), the linear equation is solved 2. If the line L and two lines y = 1, x-y-7 = 0 intersect at P and Q respectively, the middle point of the line PQ is (1, - 1) the equation for solving L
(1) When the coordinates of P point are (0,0), it means that the two intersections are symmetrical about the origin
Let the intersection of a line and the line L1: 4x + y + 6 = 0 be (m, n), then the other intersection is (- m, - n)
So it is satisfied that: 4m + N + 6 = 0 (1) - 3M + 5n-6 = 0 (2) is obtained from the simultaneous solution of (1) (2)
M = - 36 / 23, n = 6 / 23. From the two-point formula (0,0) and (- 36 / 23,6 / 23), the linear equation is
x+6y=0
(2) If the line L and two lines y = 1, x-y-7 = 0 intersect at P (x, 1) and Q (m, n), respectively,
The midpoint of PQ is (1, - 1) so n = - 3, substituting x-y-7 = 0
From the two-point equations (1, - 1) and (4, - 3), the linear equation is
2x+3y+1=0
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