Make a straight line L through point a (- 1,1) so that the midpoint of the line segment cut by two parallel lines L1: x + 2y-1 = 0 and: x + 2y-3 = 3 is exactly on the straight line L3: x-y-1 = 0, The equation for finding the line L

Make a straight line L through point a (- 1,1) so that the midpoint of the line segment cut by two parallel lines L1: x + 2y-1 = 0 and: x + 2y-3 = 3 is exactly on the straight line L3: x-y-1 = 0, The equation for finding the line L

The midpoint is on the straight line x-y-1 = 0, satisfying y = X-1. Let the midpoint be (x0, x0-1). The distance between the midpoint and two parallel lines is equal | x0 + 2 (x0-1) - 1 | / √ (1 + 4) = | x0 + 2 (x0-1) - 6 | / √ (1 + 4) | 3x0-3 | = | 3x0-8 | 3x0-3 = 8-3x0x0 = 11 / 6

Let a straight line L pass through point a (2,4), which is cut by parallel lines X-Y + 1 = 0 and x-y-1 = 0 and is the midpoint of the line segment. Then the equation of L is______ ．

The linear equation with the same distance to the parallel line X-Y + 1 = 0 and x-y-1 = 0 is X-Y = 0. The simultaneous equations x + 2Y − 3 = 0x − y = 0 are solved to x = 1y = 1. The midpoint of the line segment where l is cut by the parallel lines X-Y + 1 = 0 and x-y-1 = 0 is (1,1). The two-point equation of L is x − 12 − 1 = y − 14 − 1. That is, 3x-y-2 = 0. So the answer is: 3x-y-2 = 0

The midpoint of a line segment cut by two parallel lines X + 2Y = 1, x + 2Y = 3 is on X-Y = 1, and the angle between the line and the two parallel lines is 45 degrees

L1:x+2y=1,k(L1)=-0.5
L2:x+2y=3.k(L2)=-0.5
L3:x-y=1,k(L3)=1
Intersection a (1,0) of L1 and L3, intersection B (5 / 3,2 / 3) of L2 and L3
The midpoint m of AB (4 / 3,1 / 3)
Let the slope of the straight line l be K, then the angle between L and L1, L2 = 45 degrees. Drawing shows that there are two such lines, and the two lines are perpendicular to each other
According to the formula Tan (α - β) = (Tan α - Tan β) / (1 + Tan α · Tan β), it is obtained that
[k（L1）-k]/[1+k*k（L1）]=1
(-0.5-k)/(1-0.5k)=1
k1=-3,k2=1/3
The obtained line L:
9x+3y=13
3x-9y=1

Let a straight line L pass through point a (2,4), which is cut by parallel lines X-Y + 1 = 0 and x-y-1 = 0 and is the midpoint of the line segment. Then the equation of L is______ ．

The linear equation with the same distance to the parallel line X-Y + 1 = 0 and x-y-1 = 0 is X-Y = 0. The simultaneous equations x + 2Y − 3 = 0x − y = 0 are solved to x = 1y = 1. The midpoint of the line segment where l is cut by the parallel lines X-Y + 1 = 0 and x-y-1 = 0 is (1,1). The two-point equation of L is x − 12 − 1 = y − 14 − 1. That is, 3x-y-2 = 0. So the answer is: 3x-y-2 = 0