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=0‌2x−y=0Use 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=0‌x=1Substitute
x=1 back:
y=2x=2×1=2So, the intersection is at the point
(1,2).
Step 2: Find Another Point for Line
L2L2 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=0Subtract
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=0y=−1Substitute
y=−1 into
x−3y−2=0 :
‌x−3(−1)−2=0‌x+3−2=0‌x=−1So, the intersection point is
(−1,−1).
Step 3: Find Equation of
L2L2 passes through
(0,0) and
(−1,−1).
Slope
m=‌=1.
Using slope-intercept form:
y=x.
In general form:
−x+y=0Step 4: Find Equation of
L1L1 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=1Step 5: Find Distance Between
L1 and
L2The distance between two parallel lines
−x+y=1 and
−x+y=0 is:
Distance
=‌=‌