A curve is traced by a point $P(x,y)$ which moves such that its distance from the point $A(-1,1)$ is three times its distance from the point $B(2,-1)$. Determine the equation of the curve.
I have only one question. And that is the only thing I need answered at this time. My question to you is, when it says "which moves such that its distance from the point…" by distance, does it mean the slope from $P(x,y)$ to $A(-1,1)$ is three times the distance than from$P(x,y)$ to $B(2,-1)$? Please answer only this and nothing else. I will re-edit with further findings.
Edit:
To find the next points would it be logical to use this equation:
$d=distance$ and $P=(x,y)$
$$d(P,(-1,1))=3d(P,(2,-1))$$
$$d\sqrt{(-1\pm x_1)^2+(1\pm y_1)^2}=3d\sqrt{(2\pm x_1)^2+(-1\pm y_1)^2}$$
And from here I would use $P=(x,y)$ and plug in any values of $x$ and $y$ to try and find my equation. Would this be correct?
Best Answer
At all points on the curve the distance from $P$ to $A$ should be three times the distance from $P$ to $B$.
I recommend first sketching this by hand. Then use the distance formula to find the equation of the curve. Finally, make sure both answers agree!
Let $d(P,Q)$ represent the distance between points $P$ and $Q$. The problem statement tells us that $$d(P,A) = 3 d(P,B).$$ Plug in $(x,y)$ for $P$ and the given values of $A$ and $B$ to find the equation for the curve.
Hint: After some algebra you will find it is one of the conic sections.