# Equation for set of points

1. Sep 5, 2007

### physstudent1

1. The problem statement, all variables and given/known data

Determine the eqution for the set of all points (x,y) so that the distnce of (x,y) from (4,0) is twce the distance of (x,y) from (1,0). Describe the set geometrically.

2. Relevant equations

3. The attempt at a solution

After looking at this for a little while I figured it would be good to set the distance from the points from x,y to 4,0 equal to twice the distance from x,y to 1,0, I plugged these into the distance formula and got:

I'm kind of stuck from there and I don't know if I'm on the right track or not could someone please help. Thanks.

Last edited: Sep 6, 2007
2. Sep 5, 2007

### rock.freak667

Well that seems to be the way to do it..but maybe you'll need another equation with x and y so you can solve simultaneously and get values for x and y

here is some help: the points (1,0) and (4,0) are on the x-axis right? maybe you could just sketch the x-axis...put a point (x,y) and the points (1,0) and (4,0).. then draw the triangle formed such that the dist. of (x,y) from (1,0) is $$l$$ and the other distance is $$2l]/tex] then see if you can get another equation to help you from that 3. Sep 5, 2007 ### Dick You are off to a good start. Now square both sides of your distance equation and see if you can simplify the algebra. 4. Sep 5, 2007 ### EnumaElish That would be the case if there is only one such point. Multiple points will be described as {y = f(x) such that the distance of (x,y) from (4,0) is twice the distance of (x,y) from (1,0)}. Wouldn't it? 5. Sep 5, 2007 ### rock.freak667 well [tex]\sqrt{(4-x)^2 + (-y)^2} = 2\sqrt{(1-x)^2+(-y)^2}$$

squaring both sides and simplifying would give $$(4-x)^2+y^2=4(1-x)^2+4y^2$$
giving

$$(4-x)^2-4(1-x)^2-3y^2=0$$ which has two unknowns...either that or I keep missing something

6. Sep 5, 2007

### Dick

If you keep expanding you can see the x^2 terms will cancel too. So you can put y to be ANYTHING and get a corresponding x. There are an infinite number of solutions. The equation defines a CURVE containing an infinite number of points.

7. Sep 5, 2007

### physstudent1

I ended up getting 12 = 3x^2 +3y^2 for my equation which is half a circle.

8. Sep 5, 2007

### rock.freak667

oh..i didnt read through the question, i found the point (x,y) not describe the set geometrically

9. Sep 5, 2007

### atavistic

I think the question says find the locus and not the equation.

10. Sep 5, 2007

### physstudent1

locus? whats that..the question says to give the equation...

11. Sep 5, 2007

### rock.freak667

Doesn't the locus just describe the equation of the circle?

12. Sep 5, 2007

### Dick

That looks good. But why do you say it's only 'half' of a circle? If you can describe the curve, that is the locus.

13. Sep 5, 2007

### physstudent1

i solved it to get y to one side and graphed it on my calculator and it was a half a circle

14. Sep 5, 2007

### Dick

In solving it you took only a plus square root. The negative is also a solution.

15. Sep 5, 2007

### physstudent1

ohhh thanks a lot that is right i totally forgot about that

I am kind of confused about how I would go about getting it though when I had a square root on both sides of the equation I then squared the entire equation to simplify, would it be at this point that it would become + or - ?

Last edited: Sep 5, 2007
16. Sep 6, 2007

### Dick

You are right to be cautious, but what is inside of your radicals is always non-negative. So squaring can't introduce any false roots. It's just when you took the square root of y^2.

17. Sep 6, 2007

### HallsofIvy

Staff Emeritus
No, that's the wrong direction. squaring both sides of x= 2 gives x2= 4. If you start with x2= 4 and take the square root of both sides, then you get $x= \pm 2$

18. Sep 6, 2007

### physstudent1

So it ends up being a full circle when you have the negative root for an answer too right?
thanks a lot for the help :)

Last edited: Sep 6, 2007
19. Sep 10, 2007

### physstudent1

hey i'm just making sure that it is correct that it ends up being a circle because I heard people talking in class today and they were saying they thought it was a triangle..but I'm pretty sure that it is a circle and this is correct

20. Sep 10, 2007

### physstudent1

triangle

is this correct because I have to hand it in and people in my class were saying it should be a triangle, but I really think it should be a circle