Question

The length of a segment of a straight line through (2,3) intercepted between the straight lines y+2x =3 and y+2x=5 is 2 unit.Find the equation of the straight line.

Answer 1

L2:   y + 2x = 3                    L3:   y + 2x = 5

They are parallel lines. The difference between constant terms = 5 -3 = 2. This is the length of intercepted line segment we want.  So the desired line is either parallel or perpendicular to the x axis.

Since the line passes through  (2,3), the line is  either x = 2 or   y =3.
Intersections of both lines with line x =2,  are P(2,-1) and Q(2,1)
       Distance PQ = 2

Intersections of both lines with line y =3,  are R(0, 3) and S(1,3)
     Distance RS = 1
Answer is :  x = 2  is the straight line.

============ Another method.
   There are two answers:  x =2,  and   3x + 4 y = 18

The equation of straight line L1 passing through (2,3):
       a x + b y = c      =>          2 a + 3 b = c
  L1:   a x + b y = 2 a + 3 b
Intersection of L1 and L2 =  y + 2x = 3:
         a x + b (3 - 2x) = 2 a + 3b
   Pt P: x = 2a/(a-2b) ,     y = 3 - 2 x = (-6b-a)/(a-2b)
Intersection of L1 and L3 :  y + 2x = 5
     a x + b (5-2x) = 2a + 3b
  Pt Q:  x = (2a - 2b)/(a-2b)   ,   y = 5-2x = (a - 6b)/(a-2b)
Distance between P and Q: = 2
  so  2² = 2²b²/(a-2b)² +  2²a²/(a-2b)²
      (a-2b)² = a²+b²
      3 b² - 4 ab = 0
      b = 0  or   3b = 4 a

So equation of line :  a x = 2 a   ie.,  x = 2 
             or,    a x + 4/3 a y = 2 a + 4a 
                       ie.,  3 x + 4 y = 18

P:    4 y + 8 x = 12  => 5x = -6,  =>  4 y = 18 - 3 (-6/5) = 108/5.  
            (-6/5, 27/5)
Q:    4 y + 8 x = 20  => 5 x = 2   =>  4 y = 18 - 6/5  =84/5  
           (2/5, 21/5)
Distance PQ =   √[ (8/5)²+ (6/5)² ] = 2