Co-Ordinate Geometry - Find P for |PA - PB| is maximum

[AGNuke]

The co-ordinates of a point P on the line 2x - y + 5 = 0 such that |PA - PB| is maximum, where A is (4,-2) and B is (2, -4) will be:

A) (11, 27)
B) (-11, -17) (Answer)
C) (-11, 17) (Eliminated since it does not lie on the given line)
D) (0, 5)

Honest speaking, I don't know what to do in this question. I just attempted to prove that mirror image of A, P and B are non-collinear, but it doesn't changed the fact that I still can't reach to the answer.

Basically, I just want to ask that what is the demand of the question? What I must try to prove? Sorry for being so much of a newbie, but I can't get over with it.

I only ask for the direction in which I should work. I will try to do the rest.

Thanks!

[I like Serena]

Hi AGNuke! :smile:

Easiest is if you make a drawing of your line and your points A and B.
Now imagine picking a point P on the line and consider the difference in distances to A and B.
When you have drawn this, it should become apparent when this difference in distances is at a maximum...

[AGNuke]

and how can I derive the co-ordinates.

In the exams, we have less than 3 minutes to attempt the question and jot down the answer in the OMR sheet. That summation question I previously asked also falls in the category, despite of the fact that no one was able to solve it SUBJECTIVELY, it was done by objective approach.

You are asking me to do the same. But it would be best if I can solve it via co-ordinate geometry.

[I like Serena]

Well, in 3 minutes you can quickly make a drawing.
Hopefully you'll say "aaah, that's what they intended" and then you can immediately see what the right answer is.

In this case point P is the intersection of the specified line and the line through points A and B.

[AGNuke]

A rough sketch of the problem, I can draw it. But that's only for visualization. It is not accurate so I can't tell where my answer lies exactly.

I am just looking for a proper subjective approach.

[I like Serena]

First you need to realize that a point P for which |PA - PB| is at its maximum must lie on the line through A and B.

Then set up an equation for the line through A and B.

Solve the system of 2 equations and you have your answer.

[AGNuke]

Voila! That's quite right solution you got there. Just wondering how why this condition was necessary.

My take is for a ΔPAB, AB + PB ≥ PA
= AB ≥ PA - PB (Assuming both sides to be positive i.e. PA > PB)

Thus, maximum value of PA - PB is AB, only when the triangle is a line.

Is my explanation is right?

[I like Serena]

Voila! That's quite right solution you got there. Just wondering how why this condition was necessary.

My take is for a ΔPAB, AB + PB ≥ PA
= AB ≥ PA - PB (Assuming both sides to be positive i.e. PA > PB)

Thus, maximum value of PA - PB is AB, only when the triangle is a line.

Is my explanation is right?

Yes!
Your explanation is right.
(Although incomplete, since you've assumed PA > PB...)

[AGNuke]

Ok, I'll replace it with AB ≥ |PA - PB|

[I like Serena]

That will do it!