A point P moves so that the slope of AP is half that of BP. Where A is (0,0) and B is (0,-6). Find the equation of the locus.

- anonymous

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- anonymous

why is the point moving..lol..??

- anonymous

@surjithayer

- jim_thompson5910

let P be the point (x,y)
if A is (0,0) what is the slope of line AP ?

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## More answers

- anonymous

|dw:1443232316357:dw|

- jim_thompson5910

we'll stick with y/x
what is the slope of BP ?

- anonymous

|dw:1443232412944:dw|

- jim_thompson5910

`the slope of AP is half that of BP`
so
\[\Large \text{slope of AP} = \frac{1}{2}\times\text{slope of BP}\]
\[\Large \frac{y}{x}= \frac{1}{2}\times\frac{y+6}{x}\]
solve for y

- jim_thompson5910

surjithayer I'm not sure why you squared both sides

- anonymous

|dw:1443232684745:dw|

- jim_thompson5910

yep y = 6 is your answer

- jim_thompson5910

any point along the horizontal line y = 6 will work as long as x is not 0

- anonymous

woow really thanks

- anonymous

but y was the point moving ..lol

- anonymous

o thats not really a question... im just curious ...

- anonymous

y=-2 is not working.

- anonymous

only y=6

- jim_thompson5910

surjithayer like I said, I'm not sure why you squared both sides. You introduced an extraneous solution

- jim_thompson5910

the point moves to create the entire line

- jim_thompson5910

imagine you had this locus: the set of all points equidistant from a fixed point
that's a circle. The moving point P would circle around the center. As the point moves, it creates the locus

- anonymous

i was thinking of absolute slope but i was wrong.

- anonymous

so the locus is like the radius

- jim_thompson5910

in this case, the locus is the entire line generated by every point that makes up the line

- jim_thompson5910

each point P on this line satisfies the condition that
slope of AP = (1/2)*(slope of BP)

- jim_thompson5910

try a few points out like P = (2,6) or P = (7,6)

- anonymous

like this |dw:1443233581868:dw|

- jim_thompson5910

if P were (2,6), what's the slope of AP ?

- DanJS

interesting problem

- DanJS

since A and B have the same x-value, you can see the locus should be a horizontal line at double the vertical distance of AtoB, makes sense, in that case, both have the same change in x, and the change in y is always double for B compared to A

- DanJS

|dw:1443238354064:dw|

- DanJS

What is the relationship between the rate of change of one of the angles w.r.t. time and the rate of change of the position of P

- DanJS

maybe

- anonymous

o sorry i fall asleep

- anonymous

i review and its more clear thanks guys

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