VCAA 2007 CAS Exam 1 Q12

[joseph95]

I know this is meant to be simple but I'm clueless right now, and the answers are a little bit too summarised for my liking. Could anyone help? Thanks!

Question 12
P is the point on the line 2x + y - 10 = 0 such that the length of OP, the line segment from the origin O to P, is a minimum. Find the coordinates of P and this minimum length.

[b^3]

Let P be a point .
Now the length between any two points is given by .
So we have the points and . We can get in terms of from the equation given, as P lies on that line.

So that is the point
So the distance becomes


Now to find when this is a minimum, we differentiate it and let that equal 0.


So that is the point
The minimum length will be

EDIT: You could also minimise and work from that, as minimising it will minismise our distance function, and its less work aswell.

[BubbleWrapMan]

When I did that exam I didn't want to use the derivative of that in case I made a mistake, so I did it another way

The line given has gradient -2, so the line through O and P will have gradient 1/2 since it's perpendicular to the given line when P is closest to the origin. So the equation for the line through OP is y = x/2 or x = 2y. Substituting that into the equation for the other line gives 4y + y = 10 => y = 2 => x = 4, so P is the point (4, 2). Using pythagoras to find the distance of P from the origin gives √(16 + 4) = 2√5

[joseph95]

Thank you b^3!

And thanks ClimbTooHigh! But how did you know the line would be perpendicular? This is what confuses me with the VCAA answers.

[BubbleWrapMan]

Imagine another point, Q, on 2x + y - 10 = 0.

If we assume that OP is perpendicular to the line, then OPQ is a right-angled triangle with a right angle at P.

The hypotenuse OQ is always the longest side, and so it's longer than OP. So any line other than the perpendicular one will intersect it further from O than P, hence the perpendicular line gives the shortest distance.

[joseph95]

I'm sorry if I'm being annoying and I get the rest of your explanation, but why would you assume in the first place that the line OP is perpendicular to the original line?

Thanks heaps for this

[BubbleWrapMan]

I don't really know, it sort of just clicked to me.