1. ## Distance between lines

I 'm currently working on my exercise, the question is

Two lines
L1: 3x + y - 4 = 0 and
L2: 3x - y + 3 = 0 are given.
P (h,k) is a point on the line x - y + 1 = 0.
Suppose the distances of P from L1 and L2 are d1 and d2 respectively.

(a) Express k in terms of h.

>> I 've got it: k = h + 1

(b) Express d1 + d2 in terms of h.

>> That's easy as well.
ie. 1/(10)^(1/2) (|4h + 3| + |2h + 2|)

(c) When d1 + d2 is at the minimum, what is the value of h?
Find also the value of d1 + d2 and the coordinates of P at this instance.

>> This 's my problem. I got stuck and have no idea at all.

2. Originally Posted by ling_c_0202
I 'm currently working on my exercise, the question is
...
>> That's easy as well.
ie. 1/(10)^(1/2) (|4h + 3| + |2h + 2|)

(c) When d1 + d2 is at the minimum, what is the value of h?
Find also the value of d1 + d2 and the coordinates of P at this instance.

...

Hello,

I didn't check your final result, I can only show you how to answer the last question:

You've got a function with absolute values. Change it into a piece-wise defined function:

$|4h+3|+|2h+2|=\left\{\begin{array}{lcc}4h+3+2h+2=6 h+5,& h\geq -\frac{3}{4}\\ -4h-3+2h+2=-2h-1,& -1 \leq h < -\frac{3}{4} \\ -4h-3-2h-2=-6h-5,&h<-1 \end{array} \right.$

As you can see you've got 3 pieces of straight lines which are connected.

The change of the slope from negative to positive indicates a minimum. This happens at h = -0.75

I've attached a drawing of the graph of your function.

A Happy New Year!

EB