SOLUTION: Given A=(2,)) and B=(-1,0). Describe the set of points P satisfying the equation AP/PB=2

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Question 593105: Given A=(2,)) and B=(-1,0). Describe the set of points P satisfying the equation AP/PB=2
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
Since ")" is "lower case" 0, I'm going to assume that point A is (2, 0).

AP represents the length of segment AP (which is the distance between point A and point P. Using (x, y) for the point P and the distance formula we get:
AP =
which simplifies to:
AP =

Using the distance formula in a similar way we can find PB:
PB =
Simplifying this we get:
PB =
PB =

Substituting these expressions into the given equation AP/PB = 2 we get:

Next we will transform this equation into the general form for conic sections:

We start by squaring both sides to eliminate the square roots:

Next we will multiply each side by the denominator (to get rid of the fraction):

which gives us:

Now we just subtract from each side. (The general form is best written with a positive leading coefficient.) This gives us:

Dividing each side by 3 we get:

With A = C one would think that this would be a circle. But let's complete the squares to make sure:

We can now see that the equation is indeed the equation of a circle (with a center at (-2, 0) and a radius of 2).

We needed to complete the squares because if we had gotten:

we would have a "circle of radius 0" which is just a single point: (-2, 0).
any negative number =
we get nothing. This equation has NO solutions.