Straight Lines and Pair of Straight Lines

First, we need to find the point where the two lines x−2y+3=0 and 2x−y=0 meet.
Step 1: Find Intersection Point
To solve the two equations:
‌x−2y+3=0
‌2x−y=0
Use the second equation: 2x−y=0⟹y=2x.
Put y=2x into the first equation:
‌x−2(2x)+3=0⟹x−4x+3=0
‌−3x+3=0
‌x=1
Substitute x=1 back: y=2x=2×1=2
So, the intersection is at the point (1,2).
Step 2: Find Another Point for Line L2
L2 passes through the origin (0,0) and the intersection of 3x−y+2=0 and
x−3y−2=0.
We need the intersection point of 3x−y+2=0 and x−3y−2=0.
Multiply the second equation by 3 :
3x−9y−6=0
Subtract 3x−y+2=0 from 3x−9y−6=0 :
‌[3x−9y−6]−[3x−y+2]=0
‌(3x−9y−6)−3x+y−2=0
‌−8y−8=0
y=−1
Substitute y=−1 into x−3y−2=0 :
‌x−3(−1)−2=0
‌x+3−2=0
‌x=−1
So, the intersection point is (−1,−1).
Step 3: Find Equation of L2
L2 passes through (0,0) and (−1,−1).
Slope m=‌

−1−0

−1−0

=1.
Using slope-intercept form: y=x.
In general form: −x+y=0
Step 4: Find Equation of L1
L1 is parallel to L2 (so same slope m=1 ) and passes through ( 1,2 ).
Write y=x+c.
Plug in (1,2) :
2=1+c⟹c=1.
So Equation of L1 is y=x+1, or −x+y=1
Step 5: Find Distance Between L1 and L2
The distance between two parallel lines −x+y=1 and −x+y=0 is:
Distance =‌

|1−0|

√12+12

=‌

1

√2