Find the distance btw the two parallel lines 12x + 5y = 7 and 12x + 5y + 5 = 0 (pick any equation and let x = 0 or y = 0 and obtain the values of y and x as the case may be to get a point.)

Answer 1

Distance between two parallel line is given as d=|c₁-c₂|/sqrt(a²+b²) where two lines are ax+by+c₁=0 and ax+by+c₂=0.

Thus, d=|-7-5|/sqrt(12²+5²)=12/13

Answer 2

the 2 equations are y = (7/5) - (12/5)x and y = -1 -(12/5)x

Using the second equation to obtain a point x = 0, y = -1

Then to find the closest point on the second line will look like this, the distance formula is used

d =SQRT( (0-x)² + (-1-7/5+(12/5)x² ) gives d = SQRT((169/25)x²-(288/25)x+144/25)

Taking the derivative of the discriminant, I get (338/25)x -(288/25) and setting that to zero gives

x = .852, from which we get y = -.6448 from the first equation. Now to find the distance between them is

d = SQRT( (0-.852)² + (-1+.6448)² ) from which I obtain d = .923