Finding the Locus of a Moving Point: Solving for a Hyperbola Using Dot Products

[Appleton]

Homework Statement

A point P moves so that its distances from A(a, 0), A'(-a, 0), B(b, 0) B'(-b, 0) are related by the equation AP.PA'=BP.PB'. Show that the locus of P is a hyperbola and find the equations of its asymptotes.

Homework Equations

The Attempt at a Solution

AP.PA' = [itex]((a-x)\boldsymbol i + y\boldsymbol j).((-a-x)\boldsymbol i+y\boldsymbol j)[/itex]
AP.PA' = [itex]x^2-a^2+y^2[/itex]
BP.PB' = [itex]((b-x)\boldsymbol i + y\boldsymbol j).((-b-x)\boldsymbol i+y\boldsymbol j)[/itex]
BP.PB'= [itex]x^2-b^2+y^2[/itex]

So

[itex]a^2=b^2[/itex]

This result sugests that their is no constraint on P. This is not consistent with the question.

 

[andrewkirk]

AP.PA' = [itex]((a-x)\boldsymbol i + y\boldsymbol j).((-a-x)\boldsymbol i+y\boldsymbol j)[/itex]
AP.PA' = [itex]x^2-a^2+y^2[/itex]
BP.PB' = [itex]((b-x)\boldsymbol i + y\boldsymbol j).((-b-x)\boldsymbol i+y\boldsymbol j)[/itex]
BP.PB'= [itex]x^2-b^2+y^2[/itex]

The second line does not follow from the first line and the fourth does not follow from the third.
Write out the intermediate steps between the first and second lines and you will see what went wrong.

 

[Appleton]

Thanks for your reply andrewkirk. Unfortunately I'm still not able to identify my error. Here are the steps I omitted:

AP.PA' = [itex]((a-x)\boldsymbol i + y\boldsymbol j).((-a-x)\boldsymbol i+y\boldsymbol j)[/itex]

The dot product is distributive over vector addition, so

AP.PA' = [itex](a-x)\boldsymbol i.(-a-x)\boldsymbol i+ y\boldsymbol j.(-a-x)\boldsymbol i+(a-x)\boldsymbol i.y\boldsymbol j+y\boldsymbol j.y\boldsymbol j[/itex]

[itex]\boldsymbol i.\boldsymbol i=\boldsymbol j.\boldsymbol j=1[/itex] and [itex]\boldsymbol i.\boldsymbol j= 0[/itex]

so
AP.PA' = [itex]x^2-a^2+y^2[/itex]

Alternatively this shorthand is suggested by my book;

if
[itex]\boldsymbol a = x_1\boldsymbol i+ y_1\boldsymbol j[/itex]

and
[itex]\boldsymbol b = x_2\boldsymbol i+ y_2\boldsymbol j[/itex]

then
[itex]\boldsymbol a.\boldsymbol b=x_1x_2+y_1y_2[/itex]

so
[itex]((a-x)\boldsymbol i + y\boldsymbol j).((-a-x)\boldsymbol i+y\boldsymbol j) =(a-x)(-a-x)+y^2[/itex]

[itex]= -a^2+x^2+y^2[/itex]

 

[andrewkirk]

Actually you're quite correct. My mistake. I can't see any error in your calculations.

However, I have another idea. The question refers to the distances of P from the four points, not to the displacement vectors. If they are being careful with their words then the items AP, PA', BP, PB' are to be interpreted as scalar amounts, not vectors, and the dot between them is to be interpreted as simple multiplication (not a dot product).

I suggest trying what happens when you make that interpretation.

 

[Appleton]

Thanks for that, it makes sense now, so the asymptotes must be y =+-x