What am I doing wrong? (finding the equation of a locus)

[Joyce]

Point P moves so it is always three times as far from point B (8,0) as it is from point O (0.0). Determine the equation of the locus of P(x,y).


So, I used the distance formula and found the distance of PB and PA. I then multilpied my result for PB by three and then i grouped togather like terms. My answer was off though......


x^2+y^2 = 3x^2-48x+192+3y^2

2x^2+2y^2-48x+192=0

 

[soroban]

Hello, Joyce!

Point P moves so it is always three times as far from point B (8,0) as it is from point O (0.0).
Determine the equation of the locus of P(x,y).

You applied the \(\displaystyle 3\) to the wrong "side".

It says: "\(\displaystyle PB\) is 3 times \(\displaystyle PO\)": \(\displaystyle \,\overline{PB}\:=\:3\cdot \overline{PO}\)

So we have: \(\displaystyle \,\sqrt{(x\,-\,8)^2\,+\,y^2} \;= \;3\cdot\sqrt{x^2\,+\,y^2}\)

 

[Joyce]

o.k so I tried what you suggested but I'm still getting the wrong answer!


x^2-16x+64+y^2=3x^2+3y^2

equals: x^2+y^2+8x+8=0

The answer at the back of the book is: x^2+2x-8+y^2=0

 

[Denis]

o.k so I tried what you suggested but I'm still getting the wrong answer!
x^2-16x+64+y^2=3x^2+3y^2
equals: x^2+y^2+8x+8=0
The answer at the back of the book is: x^2+2x-8+y^2=0

your right side 3x^2 + 3y^2 should be 9x^2 + 9y^2 ; see why?

 

[Joyce]

No I don't

 

[pka]

Square both sides:
\(\displaystyle \,\sqrt{(x\,-\,8)^2\,+\,y^2} \;= \;3\cdot\sqrt{x^2\,+\,y^2}\)

 

[soroban]

Hello, Joyce!

\(\displaystyle x^2\,-\,16x\,+\,64\,+\,y^2\;=\;\)3\(\displaystyle x^2\,+\,\)3\(\displaystyle y^2\;\;\) ??

You didn't write the distances correctly in your equation . . .

The distance equation is: \(\displaystyle \,\sqrt{(x\,-\,8)^2}\;=\;3\sqrt{x^2\,+\,y^2}\)

Then: \(\displaystyle (x\,-\,8)^2\,+\,y^2\;=\;9(x^2\,+\,y^2)\)

 

[Joyce]

O i get it thanks alot everybody!