AP Calculus Circle Problem

[nolfy]

The problem is... A point (x,y) moves so that the sum of the squares of its distances from (4,1) and (2,-5) is 45. A. Show that the point moves along a circle.B. Find the center and radius.I tried the problem using several failed methods and am completely unsure how to continue. I need help getting the ball rolling. Like a concept or formula or some kind of plan of attack.Thank you in advance.

 

[srmichael]

I tried the problem using several failed methods and am completely unsure how to continue.

Show us what you HAVE tried. You may have been going down the correct path without even knowing it. It's hard for us to help you out unless we can see what youhave done and where you are stuck.

Thanks!

 

[Deleted member 4993]

The problem is... A point (x,y) moves so that the sum of the squares of its distances from (4,1) and (2,-5) is 45. A. Show that the point moves along a circle.B. Find the center and radius.I tried the problem using several failed methods and am completely unsure how to continue. I need help getting the ball rolling. Like a concept or formula or some kind of plan of attack.Thank you in advance.

What is the distance of (4,1) from (x,y)? ............ square it.........................(1)

What is the distance of (2,-5) from (x,y)? ............ square it.........................(2)

Add (1) and (2).......................................................................................(3)

Equate (3) to 45

simplify

What do you get?

 

[nolfy]

Does this mean it is an ellipse

1.(sqrt ( (x-4)^2 + (y-1)^2 ) )^2 + (sqrt ( (x-2)^2 + (y+5)^2 ) )^2=45 to........... 2. (x-4)^2 + (y-2)^2 + (x-2)^2 +(y+5)^2 = 45 .......so if that was an ellipse that equation the second one would be proof that the point (x,y) moves along a circle?

 

[pka]

1.(sqrt ( (x-4)^2 + (y-1)^2 ) )^2 + (sqrt ( (x-2)^2 + (y+5)^2 ) )^2=45 to........... 2. (x-4)^2 + (y-2)^2 + (x-2)^2 +(y+5)^2 = 45 .......so if that was an ellipse that equation the second one would be proof that the point (x,y) moves along a circle?

Do you know that:
\(\displaystyle (x-4)^2+(y-1)^2+(x-2)^2+(y+5)^2=2x^2-12x+2y^2+8y+46~?\)

 

[nolfy]

Okay yeah I factored it out and got 2x^2 -12x + 46+ 2y^2 + 8y = 45 but i am confused how do i continue, carry over the 45 or 46? then oh then it would be 2x^2-12x + 2y^2 + 8y = -1 but that doesnt make sense

 

[pka]

Okay yeah I factored it out and got 2x^2 -12x + 46+ 2y^2 + 8y = 45 but i am confused how do i continue, carry over the 45 or 46? then oh then it would be 2x^2-12x + 2y^2 + 8y = -1 but that doesnt make sense

Yes it does make sense. Complete squares.

\(\displaystyle 2(x-3)^2-18~+~2(y+2)^2-8=-1\)

 

[nolfy]

Ohhh gosh i am dumb right so that would be 2(x-3)^2 + 2(y+2)^2 = 25 so do i have to divide by two or by that my center is (3,-2) and my radius is 5 and the equation of the circle obtained is proof that point (x,y) moves along a circle right?

 

[Deleted member 4993]

Ohhh gosh i am dumb right so that would be 2(x-3)^2 + 2(y+2)^2 = 25 so do i have to divide by two or by that my center is (3,-2) and my radius is 5 and the equation of the circle obtained is proof that point (x,y) moves along a circle right?

You have to get to the form:

(y-h)² + (x-k)² = r²

For circle, those square terms on the LHS, do not have any coefficient.

So what do you need to do?

 

[nolfy]

okay so i divide by two and get (x-3)^2 + (y+2)^2= 12.5 or 25/2 so the center is (3,-2) and the radius is the sqrt of 12.5! thank you you guys are remarkable (what i said above is correct right?)

 

[nolfy]

Your right that is usually my weakness is identifing what i need to start and being confident that i am doing it right or at least on the right track. I guess it will take practice but thank you for all your guys time it made the problem and concept 10 times clearer

 

[mmm4444bot]

[r^2 = ] 25/2

the radius is the sqrt of 12.5

Your result for r is correct; yet, I would not report the radius using the expression √(12.5)

Instead, I would take the principal square root of each side of the following equation, and then rationalize the denominator.

r^2 = 25/2

Cheers :cool:

 

[HallsofIvy]

okay so i divide by two and get (x-3)^2 + (y+2)^2= 12.5 or 25/2 so the center is (3,-2) and the radius is the sqrt of 12.5! thank you you guys are remarkable (what i said above is correct right?)

Yes, that is correct. (When I first responded, I had missed that "sqrt"!)