1. Analytic last

Coordinates of A is (1,1),Coordinates of B is (5,4).
There is a point P moving over the line y=2x+4. When the sum of |AP|+|PB| is minimum with respect to P, find the ratio of PB to PA?

My work:
I found the symmetric of B. y=2x+4 so slope will be -1/2.
y-4=-1/2(x-5)
y=-x/2+13/2
y=2x+4
We find P as(1,6)

I found PA and PB AS 5 and squareroot20.

2. Re: Analytic last

Originally Posted by kastamonu
Coordinates of A is (1,1),Coordinates of B is (5,4).
There is a point P moving over the line y=2x+4. When the sum of |AP|+|PB| is minimum with respect to P, find the ratio of PB to PA?
Does the phrase "P moving over the line y=2x+4"mean that the point is on that line?
If so then the line is $\{(t,2t+4) : t\in\mathbb{R}\}$
So $|AP|+|PB|=\sqrt {{{\left( {t - 1} \right)}^2} + {{\left( {2t + 3} \right)}^2}} + \sqrt {{{\left( {t - 5} \right)}^2} + {{\left( {2t} \right)}^2}}$

SEE HERE.

3. Re: Analytic last

Yes it does.
OK.B' is (-3,8)
By using A, slope is -7/4
Equation of the line is y=-7/4x+11/4
I solved this with y=2x+4
P(-1/3,10/3)
I found PB/PA = 2.

4. Re: Analytic last

Originally Posted by kastamonu
Yes it does.
OK.B' is (-3,8)
By using A, slope is -7/4.
Slope has nothing whatsoever to do with this question.
We are asked to minimize the distance $A\text{ to }P$ plus the distance from $B\text{ to }P$: $|AP|+|PB|$
The point $A=(1,1)$, the point $P=(t,2t+4)$ so the distance $|AP|=\sqrt{(t-1)^2+(2t+4-1)^2}$.

6. Re: Analytic last

Originally Posted by kastamonu
My answer agrees with your book: $\dfrac{|AP|}{|PB|}=\dfrac{1}{2}$. SEE HERE

7. Re: Analytic last

Did you find x=-1/3 as I did?

8. Re: Analytic last

I apologise. I will check my solution again.

9. Re: Analytic last

You posted another problem yesterday which had to do with finding distances between points and there, as here, you talked about the slope of a line and finding a "symmetric" point. Neither of these problems has anything to do with either of those, just the formula for the distance between two points.

10. Re: Analytic last

Originally Posted by kastamonu
Yes it does.
OK.B' is (-3,8)
By using A, slope is -7/4
Equation of the line is y=-7/4x+11/4
I solved this with y=2x+4
P(-1/3,10/3)
I found PB/PA = 2.
Correct

B'(-3,8) is the symmetric of B with respect to the line L: y = 2x + 4

Equation of line AB': y = -7/4 x + 11/4

Point P is at the intersection of (L) and (AB'): P(-1/3,10/3)

so PB/PA = 2 or PA/PB = 1/2

Many Thanks.