Find distance of point A(2, 3) measured parallel to the line x - y = 5 from the line 2x + y + 6 = 0.

 Solution:

Given line x – y = 5 ..(i)

2x + y + 6 = 0 ..(ii)

Slope of above line is m = 1

Parallel lines have same slope.

Equation line parallel to passing through (2, 3) and slope = 1 is

y – 3 = 1(x – 2)

=> y = x – 2 + 3

=> y = x + 1 ..(iii)

Solving (iii) and (ii) we get the intersection points of line (ii) and (iii)

2x + y + 6 = 0

x – y + 1 = 0

+> 3x = -7

=> x = -7/3

=> y = x+1

= -7/3 + 1

= -4/3

So the intersection point is (-7/3, -4/3).

Distance between (2, 3) and (-7/3, -4/3) can be calculated using distance formula.

Distance = √((-7/3)-2)2 + (-4/3 – 3)2)

= √((132/9) + (132/9))

= √(169 + 169)/9

= 13√2/3 units.